r^ 


GIFT  OF 

s ton   Campbell,    Jr. 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

Microsoft  Corporation 


http://www.archive.org/details/elementsofmechanOOpeckrich 


ELEMENTS 


OF 


MEC  HAN  ICS 


TREATED     BY     MEANS    OF 


THE     DIFFERENTIAL     AND     INTEGRAL 
CALCULUS. 


BY 


WILLIAM  G.  PECK,  Ph.D.,  LL.D., 

PROFESSOB     OF     MATHEMATICS,    ASTRONOMY,    AND    MECHANICS,    COLUMBIA 
COLLEGE. 


S.    BARNES    &    COMPANY, 
NEW  YORK,  CHICAGO.  AND  NEW  ORLEANS. 


Engineer...*  C^"  V*>  N    0 

library  ?\pV 

PUBLISHERS'    NOTICE. 


PECK'S     MATHEMATICAL     SERIES. 

CONCISE,  CONSECUTIVE,  AND  COMPLETE. 


I.  FIRST    LESSONS    IN    NUMBERS. 
II.  MANUAL    OF    PRACTICAL    ARITHMETIC. 

III.  COMPLETE    ARITHMETIC. 

IV.  MANUAL    OF    ALGEBRA. 
V.  MANUAL    OF    GEOMETRY. 

VI.  TREATISE    ON    ANALYTICAL    GEOMETRY 
VII.  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 
VIII.  ELEMENTARY  MECHANICS  (without  the  Calculus). 
XL  ELEMENTS  OF  MECHANICS  (with  the  Calculus). 

Note. — Teachers  and  others,  discovering  errors  in  any  of 
the  above  works,  will  confer  a  favor  by  communicating  them 
to  us. 


Entered  according  to  Act  of  Congress,  in  the  year  1859,  by 

WILLIAM     G .    PECK, 

In  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


P  E  E  F  A  C  E 


The  following  work  was  undertaken  to  supply  a  want  felt 
by  the  author,  when  engaged  in  teaching  Natural  Philosophy 
to  College  classes.  In  selecting  a  text-book  on  the  subject 
of  Mechanics,  there  was  no  want  of  material  from  which  to 
choose ;  but  to  find  one  of  the  exact  grade  for  College 
instruction,  was  a  matter  of  much  difficulty.  The  higher 
treatises  were  found  too  difficult  to  be  read  with  profit, 
except  by  a  few  in  each  class,  in  addition  to  which  they 
were  too  extensive  to  be  studied,  even  by  the  few,  in  the 
limited  time  allotted  to  this  branch  of  education.  The 
simpler  treatises  were  found  too  elementary  for  advanced 
classes,  and  on  account  of  their  non-mathematical  character, 
not  adapted  to  prepare  the  student  for  subsequent  investi- 
gations in  Science. 

The  present  volume  was  intended  to  occupy  the  middle 
ground  between  these  two  2lasses  of  works,  and  to  form  a 
connecting  link  between  the  Elementary  and  the  Higher 
Treatises.  It  was  designed  to  embrace  all  of  the  important 
propositions  of  Elementary  Mechanics,  arranged  in  logical 
order,   and   each  rigidly  demonstrated.      If  these   designs 

834136 


IV  PREFACE. 

have  been  accomplished,  this  volume  can  be  read  with 
facility  and  advantage,  not  only  by  College  classes,  but  by 
the  higher  classes  in  Academies  and  High  Schools ;  it  will 
be  found  to  contain  a  sufficient  amount  of  information  for 
those  who  want  either  the  leisure  or  the  desire  to  make 
the  mathematical  sciences  a  specialty ;  and  finally,  it  will 
serve  as  a  suitable  introduction  to  those  higher  treatises  on 
Mechanical  Philosophy,  which  all  must  study  who  would 
appreciate  and  keep  pace  with  the  wonderful  discoveries 
that  are  daily  being  made  in  Science. 

Columbia  College,  February  22,  18ft<» 


PKEFACE    TO    THE    SECOND    EDITION. 


In  accordance  with  the  expressed  wish  of  many  teach- 
ers in  institutions  where  the  Differential  and  Integral 
Calculus  are  either  not  taught  at  all,  or  else  are  not 
obligatory  studies,  an  Appendix  has  been  added  to  the 
body  of  the  work,  in  which  all  of  the  principles  there 
demonstrated  by  means  of  the  Calculus  are  deduced  by 
the  aid  of  Elementary  Mathematics  only. 

It  has  not  seemed  desirable  to  omit  the  Calculus 
altogether,  especially  as  by  the  present  arrangement  the 
work  is  equally  adapted  to  the  use  of  those  who  teach 
by  the  aid  of  the  Calculus,  and  of  those  who  only  em- 
ploy the  Elementary  Mathematics. 

From  the  flattering  reception  of  this  work  by  the 
P  iblic,  it  is  believed  that  a  continuation  of  the  Course 
of  Natural  Philosophy,  of  which  this  is  the  opening  vol- 
ume,  would   be   acceptable.       To   carry   out   this   design, 


VI  PREFACE. 

two  other  volumes  are  in  preparation  on  the  same  gen- 
eral plan  as  the  present,  one  of  which  will  be  devoted 
to  the  subjects  of  Acoustics  and  Optics,  and  the  other 
to  Heat  and  the  Steam-Engine,  Electricity,  and  May- 
netism. 

February  22,  I860. 


CONTENTS. 


CHAPTER  I. 

FAO» 

Definitions— Rest  and  Motion 13 

Forces 14 

Gravity 15 

Weight— Mass 16 

Momentum — Properties  of  Bodies 17 

Definition  of  Mechanics — Measure  of  Forces 21 

Representation  of  Forces 23 


CHAPTER  II. 

Composition  of  Forces  whose  Directions  coincide 25 

Parallelogram  of  Forces 26 

Parallelopipedon  of  Forces 27 

Geometrical  Composition  and  Resolution  of  Forces 28 

Components  in  the  Direction  of  two  Axes 30 

Components  in  the  Direction  of  three  Axes 32 

Projection  of  Forces 34 

Composition  of  a  Group  of  Forces  in  a  Plane 35 

Composition  of  a  Group  of  Forces  in  Space 36 

Expression  for  the  Resultant  of  two  Forces 37 

Principle  of  Moments 40 

Principle  of  Virtual  Moments 43 

vii 


Vlll  CONTENTS. 

PAG* 

Resultant  of  Parallel  Forces 45 

Composition  and  Resolution — Parallel  Forces 4S 

Lever  arm  of  the  Resultant 51 

Centre  of  Parallel  Forces 52 

Resultant  of  a  Group  in  a  Plane 53 

Tendency  to  Rotation — Equilibrium  in  a  Plane 58 

Equilibrium  of  Forces  in  Space 59 

Equilibrium  of  a  Revolving  Body 60 


CHAPTER  III. 

Weight— Centre  of  Gravity 62 

Centre  of  Gravity  of  Straight  Line 64 

Of  Symmetrical  Lines  and  Areas 64 

Of  a  Triangle 65 

Of  a  Parallelogram— Of  a  Trapezoid 66 

Of  a  Polygon 67 

Of  a  Pyramid 68 

Of  Prisms,  Cylinders,  and  Polyhedrons 70 

Centre  of  Gravity  Experimentally 71 

Centre  of  Gravity  by  means  of  the  Calculus 72 

Centre  of  Gravity  of  an  Arc  of  a  Circle 73 

Of  a  Parabolic  Area 74 

Of  a  Semi-Ellipsoid 75 

Pressure  and  Stability 80 

Problems  in  Construction 85 


CHAPTER  IV. 

Definition  of  a  Machine 94 

Elementary  Machines — Cord 96 

The  Lever 98 

Tho  Compound  Lever 101 

The  Elbow-joint  Press 102 

The  Balance 103 


CONTENTS.  IX 

PAGE. 

The  Steelyard •  105 

The  Bent  Lever  Balance— Compound  Balances 106 

The  Inclined  Plane HO 

The  Pulley 112 

Single  Pulley 113 

Combinations  of  Pulleys 115 

The  Wheel  and  Axle 1 IV 

Combinations  of  Wheels  and  Axles 118 

The  Windlass   119 

The  Capstan— The  Differential  Windlass 120 

Wheel-work 121 

The  Screw 123 

The  Differential  Screw 125 

Endless  Screw 126 

The  Wedge 1 27 

General  Remarks  on  Machines 129 

Friction 130 

Limiting  Angle  of  Resistance 133 

Rolling  Friction — Adhesion 135 

Stiffness  of  Cords 136 

Atmospheric  Resistance — Friction  on  Inclined  Planes 137 

Line  of  least  Fraction 140 

Friction  on  Axle 141 


CHAPTER  V. 

Uniform  Motion 143 

Varied  Motion 144 

Uniformly  Varied  Motion 146 

Application  to  Falling  Bodies 148 

Bodies  Projected  Upwards 150 

Restrained  Vertical  Motion 153 

Atwood's  Machines 156 

Motion  on  Inclined  Planes 158 

Motion  down  a  Succession  of  Incline*  Planes 161 

Periodic  Motion 163 


X  C0NTENT8. 

tmam, 

Angular  Velocity 165 

The  Simple  Pendulum 166 

The  Compound  Pendulum 169 

Practical  Applications  of  the  Pendulum 175 

Graham's  and  Harrison's  Pendulums 176 

Basis  of  a  System  of  Weights  and  Measures 177 

Centre  of  Percussion 179 

Moment  of  Inertia 180 

Application  of  Calculus  to  Moment  of  Inertia 182 

Centre  of  Gvration 186 


CHAPTER  VI. 

Motion  of  Projectiles 18S 

Centripetal  and  Centrifugal  Forces •. .  197 

Measure  of  Centrifugal  Force 197 

Centrifugal  Force  of  Extended  Masses 203 

Principal  Axes 206 

Experimental  Illustrations 207 

Elevation  of  the  outer  rail  of  a  Curved  track 209 

The  Conical  Pendulum 210 

The  Governor 212 

Work 215 

Work,  when  the  Power  acts  obliquely '217 

Work,  when  the  Body  moves  on  a  Curve 219 

Rotation— Quantity  of  Work 225 

Accumulation  of  Work 225 

Living  Force  of  Revolving  Bodies 227 

Fly  Wheels 228 

Composition  of  Rotations 230 

Application  to  Gyroscope 232 


CHAPTER  VII. 

Classification  of  Fluids 236 

Principle  of  Equal  Pressures 236 


CONTENTS.  XI 

PAGE. 

Pressure  due  to  Weight 238 

Centre  of  Pressure  on  a  Plane  Surface 243 

Buoyant  Effect  of  Fluids 249 

Floating  Bodies ' 249 

Specific  Gravity 251 

Hydrostatic  Balance 253 

Specific  Gravity  of  an  Insoluble  Body 253 

Specific  Gravity  of  Liquids' 254 

Specific  Gravity  of  Soluble  Bodies 255 

Specific  Gravity  of  Air  and  Gases 256 

Hydrometers — Nicholson's  Hydrometer 25*7 

Scale  Areometer 258 

Volumeter 259 

Densimeter 260 

Centesimal  Alcoholometer  of  Gay  Lussac 261 

Thermometer 263 

Velocity  of  a  Liquid  through  an  Orifice 265 

Spouting  of  Liquids  on  Horizontal  Planes 268 

Modifications  due  to  Pressures 269 

Coefficients  of  Efflux  and  Velocity 270 

Efflux  through  short  Tubes 272 

Motion  of  Water  in  open  Channels 274 

Motion  of  Water  in  Pipes 277 

General  Remarks   27  9 

Capillary  Phenomena 280 

Elevation  and  Depression  between  Plates 281 

Attraction  and  Repulsion  of  Floating  Bodies 282 

Applications  of  the  principle  of  Capillarity 283 

Sndosmose  and  Exosmose 284 


CHAPTER  VIII. 

Gases  and  Vapors 286 

Atmospheric  Air 285 

Atmospheric  Pressure 286 

Mariotte's  Law 287 


Xll  CONTENTS. 

PAGE. 

Gay  Lussac's  Law 290 

Manometers — The  open  Manometer 291 

The  closed  Manometer 292 

The  Siphon  Gttage 294 

The  Barometer — Siphon  Barometer 295 

The  Cistern  Barometer 296 

Uses  of  the  Barometer 297 

Difference  of  Level 298 

Work  of  Expanding  Gas  or  Vapor 304 

Efflux  of  a  Gas  or  Vapor 306 

Steam 3"S 

Work  of  Steam 310 

Experimental  Formulas 311 


CHAPTER  IX. 

Pumps — Sucking  and  Lifting  Pumps 313 

Sucking  and  Forcing  Pump 318 

Fire  Engine 321 

The  Rotary  Pump 322 

Hydrostatic  Press 324 

The  Siphon 326 

Wurtemburg  and  Intermitting  Siphon 328 

Intermitting  Springs ; 328 

Siphon  of  Constant  Flow — Hydraulic  Ram 329 

Archimedes'  Screw. 331 

The  Chain  Pump— The  Air  Pump 332 

Artificial  Fountains— Hero's  Ball 336 

Hero's  Fountain 837 

Wine-Taster  and  Dropping  Bottle 338 

The  Atmospheric  Inkstand 838 


MECHANICS 


CHAPTEK    I. 

DEFINITIONS    AND    INTRODUCTORY    REMARKS. 

Definition  of  Natural  Philosophy.    . 

1.  Natural  Philosophy  is  that  branch  of  Science  which 
treats  of  the  laws  of  the  material  universe. 

These  laws  are  called  laws  of  nature  /  and  it  is  assumed 
that  they  are  constant,  that  is,  that  like  causes  always  pro- 
duce like  effects.  This  principle,  which  is  the  basis  of  all 
Science,  is  an  inductive  truth  founded  upon  universal  experi- 
ence. 

Definition  of  a  Body. 

2.  A  Body  is  a  collection  of  material  particles.     When 

the  dimensions  of  a  body  are  exceedingly  small,  it  is  called 

a  material  point. 

Rest  and  Motion. 

3.  A  body  is  at  rest  when  it  retains  the  same  absolute 
position  in  space ;  it  is  in  motion  when  it  continually 
changes  its  position. 

A  body  is  at  rest  with  respect  to  surrounding  objects, 
when  it  retains  the  same  relative  position  with  respect  to 
them ;  it  is  in  motion  with  respect  to  them,  when  it  con- 
tinually changes  this  relative  position.  These  states  are 
called  relative  rest  and  relative  motion,  to  distinguish  them 
from  absolute  rest  and  absolute  motion.  It  is  highly  prob- 
able that  no  object  in  the  universe  is  in  a  state  of  absolute 
rest. 


14  MECHANICS. 

Trajectory. 

4.  The  path  traced  out,  or  described  by  a  moving  point, 
is  called  its  trajectory.  When  this  trajectory  is  a  straight 
line,  the  motion  is  rectilinear  ;  when  it  is  a  curved  line,  the 
motion  is  curvilinear. 

Translation  and  Rotation. 

5.  When  all  of  the  points  of  a  body  move  in  parallel 

straight  lines,  the  motion  is  called  motion  of  translation  / 

when  the  points  of  a  body  describe  arcs  of  circles  about  a 

straight    line,   the   motion    is    called   motion    of  rotation. 

Other  varieties   of  motion   result   from   a  combination   of 

these  two. 

Uniform  and  Varied  Motion. 

6.  The  velocity  of  a  moving  point,  is  its  rate  of  motion. 

When  the  point  moves  over  equal  spaces  in  any  arbitrary 

equal    portions   of  time,  the   motion    is   uniform,  and   the 

velocity  is  constant  •  when  it  moves  over  unequal  spaces  in 

equal  portions  of  time,  the  motion  is  varied,  and  the  velocity 

is  variable.     If  the  velocity  continually  increases,  the  motion 

is   (C'cderated ;    if  it  continually  decreases,  the  motion  is 

retarded. 

Forces. 

7.  A  FoncE  is  anything  which  tends  to  change  the  state 
of  a  body  with  respect  to  rest  or  motion. 

If  a  body  is  at  rest,  anything  which  tends  to  put  it  in 
motion  is  a  force  ;  if  it  is  in  motion,  anything  which  tends 
to  make  it  move  faster,  or  slower,  is  a  force.  The  power 
with  which  a  force  acts,  is  called  its  intensity. 

Forces  are  of  two  kinds :  extraneous^  those  which  act  upon 
a  body  from  without ;  molecular,  those  which  are  exerted 
between  adjacent  particles  of  bodies. 

An  extraneous  force  may  act  for  an  instant  and  then  cease, 
in  which  ease  it  is  called  an  impulse,  or  an  impulsive  force ; 
or  it  may  act  continuously,  in  which  ease  it  is  called  an 
sant  force  An  incessant  force  may  be  regarded  as 
made  up  of  a  succession  of  impulses  acting  at  equal  but 
exceedingly  small  intervals  of  time.     When  these  successive 


DEFINITIONS    AND    INTRODUCTORY    REMARKS.  15 

impulses  are  equal,  the  force  is  constant;  when  they  are 
unequal,  the  force  is  variable.  The  force  of  gravity  at  any 
given  place,  is  an  example  of  a  constant  force ;  the  effort  of 
expanding  steam,  is  an  example  of  a  variable  force. 

Molecular  forces  are  of  two  kinds;  attractive,  those  which 
tend  to  draw  particles  together ;  repellent,  those  which  tend 
to  separate  them.  These  forces  also  exert  an  arranging 
power  by  virtue  of  which  the  particles  of  bodies  are  grouped 
into  definite  shapes.  The  phenomena  of  crystalization  pre- 
sent examples  of  this  action.  Molecular  forces  of  both  kinds 
are  continually  exerted  between  the  particles  of  all  bodies, 
and  upon  their  variation,  in  intensity  and  direction,  depend 
the  conditions  of  bodies,  whether  solid,  liquid,  or  gaseous. 

Classification  of  Bodies. 

8.  Bodies  are  divided  into  two  classes,  solids  and  fluids. 
A  solid  is  a  body  which  has  a  tendency  to  retain  a  perma- 
nent form.  The  particles  of  a  solid  adhere  to  each  other  so 
as  to  require  the  action  of  an  extraneous  force  of  greater  or 
less  intensity  to  separate  them.  A  fluid  is  a  body  whose 
particles  move  freely  amongst  each  other,  each  particle  yield- 
ing to  the  slightest  force.  Fluids  are  divided  into  liquids 
and  gases,  liquids  being  sensibly  incompressible,  whilst  gases 
are  highly  compressible.  Many  bodies  are  capable  of  exist- 
ing in  either  of  these  states  according  to  their  temperature. 
Thus  ice,  water,  and  steam,  are  simply  three  different  states 

of  the  same  body. 

Gravity. 

9.  Experiment  and  observation  have  shown  that  the  earth 
exercises  a  force  of  attraction  upon  all  bodies,  tending  to 
draw  them  towards  its  centre.  This  force,  which  is  exerted 
upon  every  particle  of  every  body,  is  called  the  force  of 
gravity.  " 

When  a  body  is  supported,  the  force  of  gravity  produces 
pressure  or  weight \  when  it  is  unsupported,  the  force  pro- 
duces motion.  Experiment  and  observation  have  shown  tl  it 
the  entire  force  of  attraction  exerted  by  the  earth  upon  any 
body,  varies  directly  as  the  quantity  of  matter  in  the  body, 


16  MECHANICS. 

and  inversely  as  the  square  of  its  distance  from  the  centre 
of  the  earth.  This  force  of  attraction  is  mutual,  so  that  the 
body  attracts  the  earth  according  to  the  same  law.  Obser- 
vation has  shown  that  this  law  of  mutual  attraction  extends 
throughout  the  universe,  and  for  this  reason  it  has  received 
the  name  of  imiversal  gravitation. 
Weight. 

10.  The  weight  of  a  body  is  the  resultant  action  of  the 
force  of  gravity  upon  all  of  its  particles.  If  the  body  there- 
fore remain  the  same,  its  weight  at  different  places  will  vary 
directly  as  the  force  of  gravity,  or  inversely  as  the  square  of 
its  distance  from  the  centre  of  the  earth. 

Mass. 

11.  The  mass  of  a  body  is  the  quantity  of  matter  which 
it  contains.  Were  the  force  of  gravity  the  same  at  every 
point  of  the  earth's  surface,  the  weight  of  a  body  might  be 
taken  as  the  measure  of  its  mass.  But  it  is  found  that  the 
force  of  gravity  increases  slightly  in  passing  from  the  equa- 
tor towards  either  pole,  and  consequently  the  weight  of  the 
same  body  increases  as  it  is  moved  from  the  equator  towards 
either  pole ;  its  mass,  however,  remains  the  same.  If  we  take 
the  weight  of  a  body  at  the  equator  as  the  measure  of  its 
mass,  it  follows  from  what  has  just  been  said,  that  the  mass 
will  be  equal  to  the  weight  at  any  place,  divided  by  the  force 
of  gravity  at  that  place,  the  force  of  gravity  at  the  equator 
being  regarded  as  the  unit ;  or,  denoting  the  mass  of  any- 
body by  3/,  its  weight  at  any  place  by  W,  and  the  force  of 
gravity  at  that  place  by  g,  we  shall  have 


W 

M  =  — ;   whence,  W  =  Mg. 


The  expression  for  the  mass  of  a  body  is  constant,  as  it 
should  be,  since  the  quantity  of  matter  remains  the  same. 

The  rxrr  of  mass  is  any  definite  mass  assumed  as  a  stand- 
ard of  comparison.     It  may  be  one  pound,  one  ounce,  or  any 


DEFINITIONS    AND    INTRODUCTORY    REMARKS.  17 

other  unit  of  weight,  taken  at  the  equator.  The  pound  is 
generally  assumed  as  the  unit  of  mass.  The  terms  weight 
and  mass  may  be  regarded  as  synonymous,  provided  we  un- 
derstand that  the  weight  is  taken  at  the  equator. 

Density. 

12.  The  density  of  a  body  is  the  quantity  of  matter 
contained  in  a  unit  of  volume  of  the  body,  or  it  is  the  mass 
of  a  unit  of  volume. 

At  the  same  place  the  densities  of  two  bodies  are  propor- 
tional to  the  weights  of  equal  volumes.  The  mass  of  any 
body  is  therefore  equal  to  its  volume  multiplied  by  its  den- 
sity, or  denoting  the  volume  by  V,  and  the  density  by  D, 
we  have 

M=   VB. 

We  have  also, 

M         W 

D   =  y  =   yr  ;    whence,    W  =    VDg. 

Momentum. 

1 3.  The  momentum  of  a  moving  body,  or  its  quantity 
of  motion,  is  the  product  obtained  by  multiplying  the  mass 
moved,  by  the  velocity  with  which  it  is  moved  ;  that  is,  we 
multiply  the  number  of  units  in  the  mass  moved  by  the  num- 
ber of  units  in  the  velocity  with  which  it  is  moved  and  the 
product  is  the  number  of  units  in  the  momentum.  This  will 
be  explained  more  in  detail  hereafter. 

Properties  of  Bodies. 

14.  All  bodies  are  endowed  with  certain  attributes,  or 
properties,  the  most  important  of  which  are,  magnitude  and 
form  /  impenetrability  ;  mobility  ;  inertia  /  divisibility,  and 
porosity ;  compressibility,  dilatibility  and  elasticity ;  at- 
traction, repulsion,  and  polarity. 

Magnitude  and  Form. 

15.  Magnitude  is  that  property  of  a  body  by  virtue  of 
which  it  occupies  a  definite  portion  of  space ;  every  body 


18  MECHANICS. 

possesses  the  three  attributes  of  extension,  length,  breadth, 
and  height.     The  form  of  a  body  is  its  figure  or  shape. 

Impenetrability. 

16.  Impenetrability  is  that  property  by  virtue  of  which 
no  two  bodies  can  occupy  the  same  space  at  the  same  time. 
The  particles  of  one  body  may  be  thrust  aside  by  those  of 
another,  as  when  a  nail  is  driven  into  wood  ;  but  where  one 
body  is,  no  other  body  can  be. 

Mobility. 

17.  Mobility  is  that  property  by  virtue  of  winch  a  body 
may  be  made  to  occupy  different  positions  at  different  in- 
stants of  time.  Since  a  body  cannot  occupy  two  positions 
at  the  same  instant,  a  certain  interval  must  elapse  whilst  the 
body  is  passing  from  one  position  to  another.  Hence  motion 
requires  time,  the  idea  of  time  bemg  very  closely  connected 

with  that  of  motion. 

Inertia. 

1§.  Inertia  is  that  property  by  virtue  of  which  a  body 
tends  to  continue  in  the  state  of  rest  or  motion  in  which  it 
may  be  placed,  until  acted  upon  by  some  force.  A  body  at 
rest  cannot  set  itself  in  motion,  nor  can  a  body  in  motion  in- 
crease or  diminish  its  rate,  or  change  the  direction  of  its  mo- 
tion. Hence,  if  a  body  is  at  rest,  it  icill  remain  at  rest,  or 
if  it  is  in  motion,  it  will  continue  to  move  uniformly  in  a 
straight  line,  until  acted  vpon  by  some  force.  This  princi- 
ple is  called  the  lav:  of  inertia.  It  follows  immediately 
from  this  law,  that  if  a  force  act  upon  a  body  in  motion,  it 
will  impart  the  same  velocity,  and  in  the  same  general  di- 
rection as  though  the  body  were  at  rest.  It  also  follows  that 
if  a  body,  free  to  move,  be  acted  upon  simultaneously  by 
two  or  more  forces  in  the  same,  or  in  different  directions,  it 
will  move  in  the  general  direction  of  each  force,  as  though 
the  other  did  not  exi^t. 

When  a  force  acts  upon  a  body  at  rest  to  produce  motion, 
or  upon  a  body  in  motion  to  change  that  motion,  a  resistance 
is  developed  equal  and  directly  opposed  to  the  effective  force 


DEFINITIONS    AND    INTRODUCTORY    REMARKS.  19 

exerted.  This  resistance,  due  to  inertia,  is  called  the  force 
of  inertia.  The  effect  of  this  resistance  is  called  re-action, 
and  the  principle  just  explained  may  be  expressed  by  saying 
that  action  and  re-action  are  equal  and  directly  opposed. 
This  principle  is  called  the  law  of  action  and  re-action. 

These  two  laws  are  deduced  from  observation  and  experi- 
ment, and  upon  them  depends  the  mathematical  theory  of 

mechanics. 

Divisibility  and  Porosity. 

19.  Divisibility  is  that  property  by  virtue  of  which  a 
body  may  be  separated  into  parts.  All  bodies  may  be  di- 
vided, and  by  successive  divisions  the  fragments  may  be  ren- 
dered very  small.  It  is  probable  that  all  bodies  are  composed 
of  ultimate  atoms  which  are  indivisible  and  indestructible  ; 
if  so,  they  must  be  exceedingly  minute.  There  are  micro- 
scopic beings  so  small  that  millions  of  them  do  not  equal  in 
bulk  a  single  grain  of  sand,  and  yet  these  animalcules  possess 
organs,  blood,  and  the  like.  How  inconceivably  minute,  then, 
must  be  the  atoms  of  which  these  various  parts  are  composed. 

Porosity  is  that  property  by  virtue  of  which  the  particles 
of  a  body  are  more  or  less  separated.  The  intermediate 
spaces  are  called  pores.  When  the  pores  are  small,  the  body 
is  said  to  be  dense ;  when  they  are  large,  it  is  said  to  be  rare. 
Gold  is  a  dense  body,  air  or  steam  a  rare  one. 

Compressibility,  Dilatability,  and  Elasticity. 

20.  Compressibility,  or  contractility,  is  that  property  by 
virtue  of  which  the  particles  of  a  body  are  susceptible  of 
being  brought  nearer  together,  and  dilatability  is  that  prop- 
erty by  virtue  of  which  they  may  be  separated  to  a  greater 
distance.  All  bodies  contract  and  expand  Avhen  their  tem- 
peratures are  changed.  Atmospheric  air  is  an  example  of 
a  body  which  readily  contracts  and  expands. 

Elasticity  is  that  property  by  virtue  of  which  a  body  tend* 
to  resume  its  original  form  after  compression,  or  extension. 
Steel  and  India  rubber  are  instances  of  elastic  bodies.  No 
bodies  are  perfectly  elastic,  nor  are  any  perfectly  inelastic. 
The  force  which  a  body  exerts  in  endeavoring  to  resume  its 


20  MECHANICS. 

form  after  distortion,  is  called  the  force  of  restitution.  If 
we  denote  the  force  of  distortion  by  <#,  the  force  of  restitu- 
tion by  r,  and  their  ratio  by  e,  we  shall  have 

r 

in  which  e  is  called  the  modulus  of  elasticity.  Those 
bodies  are  most  elastic  which  give  the  greatest  value  for  e. 
Glass  is  highly  elastic,  clay  is  very  inelastic. 

Attraction,  Repulsion,  and  Polarity. 

2 1 .  Attraction  is  that  property  by  virtue  of  which  one  par- 
ticle has  a  tendency  to  pull  others  towards  it.  Repulsion  is 
that  property  by  virtue  of  which  one  particle  tends  to  push 
others  from  it.  The  dissimilar  poles  of  two  magnets  attract 
each  other,  whilst  similar  poles  repel  each  other.  It  is  sup- 
posed that  forces  of  attraction  and  repulsion  are  continually 
exerted  between  the  neighboring  particles  of  bodies,  and  that 
the  positions  of  these  particles  are  continually  changing,  as 
these  forces  vary. 

Polarity  is  that  property  by  virtue  of  which  the  attractive 
and  repellent  forces  between  the  particles  exert  an  arranging 
power,  so  as  to  give  definite  forms  to  masses.  The  phenom- 
ena of  crystalization  already  referred  to,  depend  upon  this 
property.  It  is  to  polarity  that  many  of  the  most  interest- 
ing phenomena  of  physics  are  to  be  attributed. 

Equilibrium. 

22.  A  system  of  forces  is  said  to  be  in  equilibrium  when 
they  mutually  counteract  each  other's  effects.  If  a  system 
of  forces  in  equilibrium  be  applied  to  a  body,  they  will  not 
change  its  state  with  respect  to  rest  or  motion  ;  if  the  body 
be  at  rest  it  will  remain  so,  or  if  it  be  in  motion,  it  will  con- 
tinue to  move  uniformly,  so  far  as  these  forces  are  concerned. 
The  idea  of  an  equilibrium  of  forces  does  not  imply  either 
rest  or  motion,  but  simply  a  continuance  in  the  previous 
state,  with  respect  to  rest  or  motion.  Hence  two  kinds  of 
equilibrium  are  recognized ;  the  equilibrium  of  rest,  called 


DEFINITIONS    AND    INTRODUCTORY    REMARKS.  21 

statical  equilibrium,  and  the  equilibrium  of  motion,  called 
dynamical  equilibrium.  If  we  observe  that  a  body  remains 
at  rest,  we  infer  that  all  the  forces  acting  upon  it  are  in  equi- 
librium ;  if  we  observe  that  a  body  moves  uniformly,  we  in 
like  manner  infer  that  all  the  forces  acting  upon  it  are  in 
equilibrium. 

Definition  of  Mechanics. 

23.  Mechanics  is  that  science  which  treats  of  the  laws 
of  equilibrium  and  motion.  That  branch  of  it  which  treats 
of  the  laws  of  equilibrium  is  called  statics ;  that  branch 
which  treats  of  the  laws  of  motion  is  call  ed  dynamics.  When 
the  bodies  considered  are  liquids,  of  which  water  is  a  type, 
these  two  branches  are  called  hydrostatics  and  hydrodynam- 
ics. When  the  bodies  considered  are  gases,  of  which  air  is 
a  type,  these  brandies  are  called  aerostatics  and  aerody- 
namics. 

Measure  of  Forces. 

24.  We  know  nothing  of  the  absolute  nature  of  forces, 
and  can  only  judge  of  them  by  their  effects.  We  may,  how- 
ever, compare  these  effects,  and  in  so  doing,  we  virtually 
compare  the  forces  themselves.  Forces  may  act  to  produce 
pressure,  or  to  produce  motion.  In  the  former  case,  they 
are  called  forces  of  pressure  y  in  the  latter  case,  moving 
forces.  There  are  two  corresponding  methods  of  measuring 
forces,  first,  by  the  pressure  they  can  exert,  secondly,  by  the 
quantities  of  motion  which  they  can  communicate. 

A  force  of  pressure  may  be  expressed  in  pounds;  thus,  a 
pressure  of  one  pound  is  a  force  which,  if  directed  vertically 
upwards,  would  just  sustain  a  weight  of  one  pound  ;  a  pres- 
sure of  two  pounds  is  a  force  which  would  sustain  a  weight 
of  two  pounds,  and  so  on. 

A  moving  force  may  be  a  single  impulse,  or  it  may  be 
made  up  of  a  succession  of  impulses. 

The  unit  of  an  impulsive  force,  is  an  impulse  which  can 
cause  a  unit  of  mass  to  move  over  a  unit  of  space  in  a  unit 
of  time.  A  force  which  can  cause  two  units  of  mass  to  move 
over  a  unit  of  space  in  a  unit  of  time,  or  which  can  cause  a 


22  MECHANICS. 

unit  of  mass  to  move  over  two  units  of  space  in  a  unit  of 
time,  is  called  a  double  force. 

A  force  which  can  cause  three  units  of  mass  to  move  over 
a  unit  of  space  in  a  unit  of  time,  or  which  can  cause  a  unit 
of  mass  to  move  over  three  units  of  space  in  a  unit  of  time, 
is  called  a  triple  force,  and  so  on. 

If  we  represent  a  unit  of  force  by  1,  a  double  force  will 
be  represented  by  2,  a  triple  force  by  3,  and  so  on. 

In  general,  a  force  which  can  cause  m  units  of  mass  to 
move  over  n  units  of  space  in  a  unit  of  time,  will  be  repre- 
sented by  m  x  n.  Hence,  forces  may  be  compared  with 
each  other  as  readily  as  numbers,  and  by  the  same  general 
rules. 

The  unit  of  mass,  the  unit  of  space,  and  the  unit  of  time, 
are  altogether  arbitrary,  but  having  been  once  assumed  they 
must  remain  the  same  throughout  the  same  discussion.  We 
shall  assume  a  mass  weighing  one  pound  at  the  equator,  as 
the  unit  of  mass,  one  foot,  as  the  unit  of  space,  and  one 
second,  as  the  unit  of  time. 

Let  us  denote  any  impulsive  force,  by  f  the  mass  moved, 
by  m,  and  the  velocity  which  the  impulse  can  impart  to  it  by 
v.  Then,  since  the  velocity  is  the  space  passed  over  in  one 
second,  we  shall  have,  from  what  precedes, 

f  —  mv. 

If  we  suppose  m  to  be  equal  to  1,  we  shall  have, 

/=    V. 

That  is,  the  measure  of  an  impulse  is  the  velocity  which  it 
ca?i  impart  to  a  unit  of  mass. 

An  incessant  force  is  made  of  a  succession  of  impulses.  It 
has  been  agreed  to  take,  as  the  measure  of  an  incessant  force, 
the  quantity  of  motion  that  it  can  generate  in  one  second,  or 
the  unit  of  time. 

If  we  denote  an  incessant  force  by  /,  the  mass  moved  by 
m,  and  the  velocity  generated  in  one  second  by  v,  we  shall 
have, 

/  =  mv. 


DEFINITIONS    AND    INTRODUCTORY    REMARKS.  23 

If  we  suppose  m  to  be  equal  to  1,  we  shall  have, 

/  =  ». 

That  is,  the  measure  of  an  incessant  force  is  the  velocity 
which  it  can  generate  in  a  unit  of  mass  in  a  unit  of  time. 
If  the  force  is  of  such  a  nature  as  to  act  equally  upon 
every  particle  of  a  body,  as  gravity,  for  instance,  the  vel- 
ocity generated  will  be  entirely  independent  of  the  mass. 
In  these  cases,  the  velocity  that  a  force  can  generate  in  a  unit 
of  time,  is  called  the  acceleration  due  to  the  force.  If  we 
denote  the  acceleration  hjf  the  mass  acted  upon  by  m,  and 
the  entire  moving  force  hyf,  we  shall  have, 

f  ~  mf  =  mv. 

Since  an  incessant  force  is  made  up  of  a  succession  of  im- 
pulses, its  measure  may  be  assimilated  to  that  of  an  impul- 
sive force,  so  that  both  may  be  represented  and  treated  in 
the  same  manner. 

Forces  of  pressure,  if  not  counteracted,  would  produce 
motion ;  and,  as  they  differ  in  no  other  respect  from  the 
forces  already  considered,  they  also  may  be  assimilated  to 
impulsive  forces,  and  treated  in  the  same  manner. 

Representation  of  Forces. 

25.  It  has  been  found  convenient  in  Mechanics  to  repre- 
sent forces  by  straight  lines;  this  is  readily  effected  by 
taking  lines  proportional  to  the  forces  which  they  repre- 
sent. Having  assumed  some  definite  straight  line  to  repre- 
sent a  unit  of  force,  a  double  force  will  be  represented  by  a 
line  twice  as  long,  a  triple  force  by  a  line  three  times  as  long, 
and  so  on. 

A  force  is  completely  given  when  we  have  its  intensity, 
its  point  of  application,  and  the  direction  in  which  it  acts. 
When  a  force  is  represented  by  a  straight  line,  the  length  of 
the    line    represents    the    intensity,    one 

extremity  of  the  line  represents  the  point     » *~ 

of  application,  and  the  direction  of  the  pig.  1. 

line  represents  the  direction  of  the  force. 

Thus,  in  figure  1,  0  P  represents  the  intensity,  0  the  point 


24 


MECHANICS. 


of  application,  and  the  direction  from  0  to  P  is  the  direction 
of  the  force.     This  direction  is  gen- 
erally indicated  by  an  arrow  head.  g ^ 

It  is  to  be  observed  that  the  point  of  Fig.  1. 

application  of  a  force  may  be  taken 

at  any  point  of  its  line  of  direction,  and  it  is  often  found 

convenient  to  transfer  it  from  one  point  to  another  on  this 

line. 

The  intensity  of  a  force  may  be  represented  analytically 
by  a  letter,  which  letter  is  usually  the  one  placed  at  the  ar- 
row head ;  thus,  in  the  example  just  given,  we  should  desig- 
nate the  force  OP  by  the  single  letter  P. 

If  forces  acting  in  any  direction  are  regarded  as  positive, 
those  acting  in  a  contrary  direction  must  be  regarded  as  nega- 
tive. This  convention  enables  us  to  apply  the  ordinary  rules 
of  analysis  to  the  investigations  of  Mechanics. 

Forces  situated  in  the  same  plane  are  generally  referred  to 
two  rectangular  axes,  OX  and  0  Y, 
which  are  called  co-ordinate  axes. 
The  direction  from  0  towards  JTis 
that  of  positive  abscissas  ;  that  from 
0  towards  X  is  that  of  negative  ab- 
scissas. The  directions  from  0  to- 
wards Y  and  Y\  respectively,  are 
those  of  positive  and  negative  ordi- 
nates.  Forces  acting  in  the  direc- 
tions of  positive  abscissas  and  posi- 
tive ordinates  are  positive ;  those 
acting  in  contrary  directions,  are 
negative. 

Forces  in  space  are  referred  to 
three  rectangular  co-ordinate  axes, 
OX,  0  Y,  and  OZ.  Forces  acting 
from  0  towards  X,  Y,  or  Z,  are 
positive,  those  acting  in  contrary 
directions,  are  negative.  Fi{ 


IY7 
Fig.  2. 


IZ 


.   3. 


COMPOSITION    AND   RESOLUTION    OF   FORCES.  25 


CHAPTER    II. 

COMPOSITION,  RESOLUTION,  AND  EQUILIBRIUM  OF  FORCES. 

Composition  of  Forces  whose  directions  coincide. 

26.  Composition  of  forces,  is  the  operation  of  finding  a 
single  force  whose  effect  is  equivalent  to  that  of  two  or  more 
given  forces.  This  single  force  is  called  the  resultant  of  the 
given  forces.  Resolution  of  forces,  is  the  operation  of  find- 
ing two  or  more  forces  whose  united  effect  is  equivalent  to 
that  of  a  given  force.  These  forces  are  called  components 
of  the  given  force. 

If  two  forces  are  applied  at  the  same  point,  and  act  in  the 
same  direction,  their  resultant  is  equal  to  the  sum  of  the  two 
forces.  If  they  act  in  contrary  directions,  their  resultant  is 
equal  to  their  difference,  and  acts  in  the  direction  of  the 
greater  one.  In  general,  if  any  number  of  forces  are  ap- 
plied at  the  same  point,  some  of  which  act  in  one  direction, 
and  the  others  in  a  contrary  direction,  their  resultant  i? 
equal  to  the  sum  of  those  which  act  in  one  direction,  dimin- 
ished by  that  of  those  which  act  in  the  contrary  direction  • 
or,  if  we  regard  the  rule  for  signs,  the  resultant  is  equal  to 
the  algebraic  sum  of  the  components  ;  the  sign  of  this  alge- 
braic sum  makes  known  the  direction  in  which  the  resultant 
acts.  This  principle  follows  immediately  from  the  rule 
adopted  for  measuring  forces. 

Thus,  if  the  forces  P,  P',  &c,  applied  at  any  point,  act  in 
the  direction  of  positive  abscissas,  whilst  the  forces  P",  P'", 
&c,  applied  to  the  same  point,  act  in  the  direction  of  nega- 
tive abscissas,  then  will  their  resultant,  denoted  by  P,  be 
given  by  the  equation, 

R  =  (P  +  P'  +  etc.,)  -  (P"  +  P"  +  &c.) 

2 


26  MECHANICS. 

If  the  first  term  of  the  second  member  of  this  equation  is 
numerically  greater  than  the  second,  P  is  positive,  which 
shows  that  the  resultant  acts  in  the  direction  of  positive  ab- 
scissas. If  the  first  term  is  numerically  less  than  the  second, 
P  is  negative,  which  shows  that  the  resultant  acts  in  the 
direction  of  negative  abscissas. 

If  the  two  terms  of  the  second  member  are  numerically 
equal,  P  will  reduce  to  0.  In  this  case,  the  forces  will  exact- 
ly counterbalance  each  other,  and,  consequently,  will  be  in 
equilibrium. 

Whenever  a  system  of  forces  is  in  equilibrium,  their  re- 
sultant must  necessarily  be  equal  to  0.  "When  all  of  the 
forces  of  the  system  are  applied  at  the  same  point,  this  sin- 
gle condition  will  be  sufficient  to  determine  an  equilibrium. 

All  of  the  forces  of  a  system  which  act  in  the  general  di- 
rection of  the  same  straight  line,  are  called  homologous,  and 
their  algebraic  sum  may  be  expressed  by  writing  the  ex- 
pression for  a  single  force,  prefixing  the  symbol  2,  a  sym- 
bol which  indicates  the  algebraic  sum  of  several  homologous 
quantities.  We  might,  for  example,  write  the  preceding 
equation  under  the  form, 

R  =  *{P) (1.) 

This  equation  expresses  the  fact,  that  the  resultant  of  a  sys- 
tem of  forces,  acting  in  the  same  direction,  is  equal  to  the 
algebraic  sum  of  the  forces. 

Parallelogram  of  Forces. 

27.  Let  P  and  Q  be  two  forces  applied  to  the  material 
point  0,  taken  as  a  unit  of  mass,  and 

acting  in  the  directions  OP  and   OQ.  q  _  "R 

Let  OP  represent  the  velocity  gener-  /  ^i 

ated  by  the  force  P,  and  OQ  the  ve-         /        /^ 
locity  generated  by  the  force  Q.   Draw       Z^___— -w' 
PP  parallel  to  OQ,  and  QE  parallel     0 
to  OP  ;  draw  also  the  diagonal  OP.  ***•  4 

From  the  law  of  inertia  (Art  18),  it  follows  that  a  mass 
acted  upon  by  two  simultaneous  forces  moves  in  the  general 


COMPOSITION    AND   RESOLUTION    OF   FORCES.  '2T 

direction  of  each,  as  though  the  other  did  not  exist.  Now, 
if  we  suppose  the  material  point  0,  to  be  acted  upon  simul- 
taneously by  the  two  forces  P  and  Q,  it  will,  by  virtue  of  the 
first,  be  found  at  the  end  of  one  second  somewhere  on  the 
line  PP ;  and  by  virtue  of  the  second  somewhere  on  the 
line  Qll ;  hence,  it  will  be  at  their  point  of  intersection. 
But  had  the  point  0  been  acted  upon  by  a  single  force,  rep- 
resented in.  direction  and  intensity  by  OP,  it  would  have 
moved  from  0  to  P  in  the  same  time.  Hence,  the  single 
force  P  is  equivalent,  in  effect,  to  the  aggregate  of  the  two 
forces  P  and  Q  ;  it  is,  therefore,  their  resultant.     Hence, 

If  two  forces  be  represented  in  direction  and  intensity  by 
the  adjacent  sides  of  a  parallelogram,  their  resultant  tcill  be 
represented  in  direction  and  intensity  by  that  diagonal  of 
the  parallelogram,  which  jxisses  through  their  point  of  in- 
tersection. 

This  principle  is  called  the  parallelogram  of  forces. 

In  the  preceding  demonstration  we  have  only  considered 
moving  forces,  but  the  principle  is  equally  true  for  forces  of 
pressure ;  for,  if  we  suppose  a  force  equal  and  directly  op- 
posed to  the  resultant  P,  this  force  will  be  in  equilibrium 
with  the  forces  P  and  Q,  which  will  then  become  forces  of 
pressure.  The  relation  between  the  forces  will  not  be 
changed  by  this  hypothesis,  and  we  may  therefore  enunciate 
the  principle  as  follows : 

If  two  pressures  be  represented  in  direction  and  intensity 
by  the  adjacent  sides  of  a  parcdlelogrcmi,  their  resultant 
icill  be  rep/resented  in  direction  and  intensity  by  that  diago- 
nal of  the  parallelogram  which  passes  through  their  com 
mon  point. 

This  principle  is  called  the  parallelogram  of  pressures. 

Hence,  we  see  that  moving  forces  and  pressw-es  may  be 
compounded  and  resolved  according  to  the  same  principles, 
and  by  the  same  general  laws. 

Parallelopipedon  of  Forces. 

28.  Let  P,  Q,  and  S  represent  three  forces  applied  to 
the  same  point,  and  not  in  the  same  plane.    Upon  these  lines, 


28 


MECHANICS. 


as  ed&is*  ,  onstruct  the  parallelopipedon  OH,  and  draw  0JS1 
and  Ssi.     From  the  preceding  article, 
OM  represents  the  resultant  of  P  and 
Q,  and  from  the  same  article,  OH  rep- 
resents the  resultant  of  031  and  S. 

Hence,  OH  is  the  resultant  of  the 
three  forces  P,  Q,  and  S.  That  is,  if 
three  forces  be  represented  in  direc- 
tion and  intensity  by  three  adjacent 

edges  of  a  parallelopipedon,  their  resultant  will  be  repre- 
sented by  that  diagonal  of  the  p>arcdlelopipedon  which 
passes  through  their  point  of  intersection. 

This  principle  is  known  as  the  parallelopipedon  of  forces, 
and  is  equally  true  for  moving  forces  and  pressures. 

Geometrical  Composition  and  Resolution  of  Forces. 

29.  The  following  constructions  depend  upon  the  prin- 
ciple of  the  parallelogram  of  forces. 

1.  Having  given  the  directions  and  intensities  of  two 
forces  applied  at  the  same  point,  to  find  the  direction  and  in- 
tensity of  their  resultant. 

Let  OP  and  OQ  represent  the 
given  forces,  and  0  their  point  of  ap- 
plication; draw  PH  parallel  to  OQ, 
and  QH  parallel  to  OP,  and  draw 
the  diagonal  OH ;  it  will  be  the  re- 
sultant sought. 

2.  Having  given  the  direction  and  intensity  of  the  result- 
ant of  two  forces,  and  the  direction  and  intensity  of  one  of 
its  components,  to  find  the  direction  and  intensity  of  the 
other  component. 

Let  H  be  the  given  resultant,  P  the  given  component,  and 
0  their  point  of  application  ;  drawiiP,  and  through  0  draw 
OQ  parallel  to  HP,  also  through  H  draw  HQ  parallel  to 
P  0\  then  will  OQ  be  the  component  sought. 

3.  Having  given  the  direction  and  intensity  of  the  result- 
ant of  two  forces,  and  the  directions  of  the  two  components, 
to  find  the  intensities  of  the  components. 


T\z.  6. 


COMPOSITION    AND    RESOLUTION    OF    FORCES. 


29 


Let  P  be  the  given  resultant,  OP 
and  OQ  the  directions  of  the  compo- 
nents, and  0  their  point  of  applica- 
tion. Through  P  draw  PP  and  PQ 
respectively,  parallel  to  Q  0  and  P  0, 
then  will  OP  and  OQ  represent  the  intensities  of  the  com- 
ponents. 

From  this  construction  it  is  evident  that  any  force  may 
be  resolved  into  two  components  having  any  direction  what- 
ever ;  these,  again  may  each  be  resolved  into  new  compo- 
nents, and  so  on ;  hence  it  follows  that  a  single  force  may  be 
resolved  into  any  number  of  components  having  any  as- 
sumed directions  whatever. 

4.  Having  given  the  direction  and  intensity  of  the  re- 
sultant of  two  forces,  and  the  intensities  of  the  components, 
to  find  their  directions. 

Let  P  be  the  given  resultant,  and 
0  its  point  of  application.  With  P 
as  a  centre,  and  one  of  the  compo- 
nents as  a  radius,  describe  an  arc  of 
a  circle ;  with  0  as  a  centre,  and  the 
other  component  as  a  radius,  describe 

a  second  arc  cutting  the  first  at  P ;  draw  PP  and  P  0,  and 
complete  the  parallelogram  PQ,  then  will  OP  and  OQ  be 
the  directions  sought. 

5.  To  find  the  resultant  of  any  number  of  forces,  P,  Q, 
JSt  T,  &c,  lying  in  the  same  plane,  and  applied  at  the  same 
point.  Construct  the  resultant  P' 
of  P  and  Q,  then  construct  the  re- 
sultant P"  of  P'  and  £,  then  the 
resultant  P  of  P"  and  T,  and  so  on: 
the  final  resultant  will  be  the  result- 
ant of  the  system. 

By  inspecting  the  preceding  fig- 
ure, we  see  that  in  the  polygon  OQ 
P'P"PT,  the  side  QP'  is  equal  and 
parallel  to  the  force  P,  the  side 
R'P"  to  the  force  JS,  and  the  side  P"P  to  the  force   T, 


j  V. 

V 


Fig.  9. 


30  MECHANICS. 

and  so  on.  Hence,  we  may  construct  the  resultant  of  such 
a  system  of  forces  by  drawing  through  the  second  extremity 
of  the  first  force,  a  line  parallel  and  equal  to  the  second 
force,  through  the  second  extremity  of  this  line,  a  line  par- 
allel and  equal  to  the  third  force,  and  so  on  to  the  last.  The 
line  drawn  from  the  starting  point  to  the  last  extremity  of 
the  last  line  drawn,  will  represent  the  resultant  sought.  If 
the  last  extremity  of  the  last  force  fall  at  the  starting  point, 
the  resultant  will  be  0,  and  the  system  will  be  in  equili- 
brium. 

This  principle  is  called  the  polygon  of  forces  ;  its  simplest 
case  is  the  triangle  of  forces. 

Components  of  a  Force  in  the  direction  of  two  axes. 

30.     To  find  expressions  for  the  components  of  a  force 
which  act  in  directions  parallel  to  two 
rectangular  axes.     Let  OX  and  OY  be  Yjr r 

two  such  axes,  and  II  any  force  lying 
in   their  plane;    construct   the   compo- 
nents parallel  to  OX  and  0  Y,  as  be-     - 
fore    explained,  and  denote   the  angle 
LAB.,  which  the  force  makes  with  the  ,,.    1f, 

r  lg.  10. 

axis  of  JT,  by  a.     From  the  figure,  we 
have, 

AL  =  R  cos  a,  and  BL  —  AM  —  II  sin  a  ; 

or,  making  AL  =  X,  and  A 31  r=  Y,  we  have, 

X  =  B  cos  a,  and  Y  =  II  sin  a    .     .     (2.) 

The  angle  a  is  estimated  from  the  direction  of  positive 
abscissas  around  to  the  left  through  360°. 

For  all  values  of  a  from  0°  to  90°,  and  from  270°  to  360°, 
the  cosine  of  a  will  be  positive,  and,  consequently,  the  com- 
ponent  ylXwill  be  positive;  that  is,  it  will  act  in  the  direction 
of  positive  abscissas.  For  all  values  of  a  from  90°  to  270°, 
the  cosine  of  «  will  be  negative,  and  the  component  AL 
will  act  in  the  direction  of  negative  abscissas. 


COMPOSITION    AND    RESOLUTION    OF    FORCES. 


3i 


Pig.  10. 


For  all  values  of  a  from  0°  to  180°,  the  sine  of  a  will  be 
positive,  and  the  component  AM  will 
be  positive  ;  that  is,  it  will  act  in  the 
direction  of  positive  ordinates.  For  all 
values  of  a  from  180°  to  360°,  the  sine 
of  a  will  be  negative,  and  the  compo-  "~ 
nent  A3I  will  act  in  the  direction  of 
negative  ordinates. 

For  a  =   90°,  or  a  =  270°,  we  shall 
have  AL  =  0.     For  a  =   0,  or  a  =   180°,  we  shall  have 
AM  =  o. 

If  we  regard  AL  and  A3fns  two  given  forces,  R  will  be 
their  resultant ;  and  since  RL  =  A3f,  we  shall  have  from 
the  figure, 

R  =  ^Tx^r-T*     ....    (3.) 

Hence,  the  resultant  of  any  two  forces,  at  right-angles  to 
each  other,  is  equal  to  the  square  root  of  the  sum  of  the 
squares  of  the  two  forces. 

From  the  figure,  we  also  have, 


cos  a 


R' 


and 


sin  a  =  -= 


Y 
R 


Hence,  the  resultant  is  completely  determined. 

PRACTICAL      EXAMPLES.. 

1.  Two  pressures  of  9  and  12  pounds,  respectively,  act 
upon  a  point,  and  at  right-angles  to  each  other.  Required, 
the  direction  and  intensity  of  the  resultant  pressure. 


We  have, 

X=  9,   and  T=  12; 

5 -'i 


SOLUTION. 


Also,     COS  a 


R=  V81~+  144  -  15. 
a  =  53°  r  47." 


That  is,  the  resultant  pressure  is  15  lbs.,  and  it  makes  an 
angle  of  53°  7'  47"  with  the  direction  of  the  first  force. 
2.     Two  forces  are  to  each  other  as  3  is  to  4,  and  their 


32  MECHANICS. 

resultant  is  20  lbs.     What  are  the  intensities  of  the  compo- 
nents ? 

SOLUTION. 

We  have,    3Y=4X,   or    F=   f-XJ  and  R  =  20; 

.-.     20  =  y^X2  +  ^X*  =  f  A7"; 

Hence,  JT  =  12,    and  F=  16. 

3.  A  boat  fastened  by  a  rope  to  a  point  on  the  shore,  is 
urged  by  the  wind  perpendicular  to  the  current,  with  a  force 
of  18  pounds,  and  down  the  current  by  a  force  of  22  pounds. 
What  is  the  tension,  or  strain,  upon  the  rope,  and  what 
angle  does  it  make  with  the  current  ? 


SOLUTION. 

We  have 

X  =  22,    and  T  =   18  ;  .'.     R  =  V808  =  28.425  ; 


Also,      cos  a  = 


99 


28.425 


39°  17'  20". 


Hence   the   tension   is   28.425  lbs.,    and   the    angle    39° 
W  20". 

Components  of  a  Force  in  the  direction  of  three  axes. 
31.  To  find  expressions  for  the  components  of  a  force  ia 
the  directions  of  three  rectangu- 
lar axes.  Let  OR  represent  the 
force,  and  OX,  OT,  and  OZ, 
three  rectangular  axes  drawn 
through  its  point  of  application, 

O.  Construct  a  parallelopipedon 
on  OR  as  a  diagonal,  having 
three  of  its  edges  coinciding  with 
the    axes.      Then    will   the   lines 

OX,    OM,    and    OX,    represent 

the  required   components.     Denote   these  components,  re- 
spectively, by  X,  Y,  and  Z.  Draw  lines  from  R,  to  X,  31,  and 


7r* 


/r 


Fie.  11. 


COMPOSITION    AND    RESOLUTION    OF    FORCES. 


33 


3rx 


Fig.  1L 


N,  respectively ;  these  will  be  perpendicular  to  the  axes,  and 
with  them,  and  the  force  i?, 
will  form  three  right-angled 
triangles.  Denote  the  angle 
between  JR  and  the  axis  of  JC 
by  a,  that  between  H  and  the 
axis  of  y  by  /3,  and  that  between 
11  and  the  axis  of  Z  by  7 ;  we 
shall  have  from  the  right-angled 
triangles  referred  to,  the  follow- 
ing equations : 

JC  =  JR  cos  a,    Y  =  R  cos  /3,   and  Z  =  R  cos  7. 

The  angles  a,  /3,  and  7,  are  estimated  from  the  directions 
of  the  positive  co-ordinates,  through  360°.  The  components 
above  found  will  be  positive  when  they  act  in  the  direction 
of  positive  co-ordinates,  and  negative  when  they  act  in  a 
contrary  direction. 

If  we  regard  J5T,  Y,  and  Z,  as  three  forces,  R  will  be 
their  resultant,  and  we  shall  have,  from  a  known  property 
of  the  rectangular  parallelopipedon, 


R  =  ^X%  +  Y*  +  Z* 


(4.) 


That  is,  the  resultant  of  three  forces  at  right  angles  to 
each  other,  is  equal  to  the  square  root  of  the  sum  of  the 
squares  of  the  components. 

We  also  have  from  the  figure, 


cos 


X  a         Y       a  Z 

a,  =  — ,  cos  ,0   =  -^,  and  cos  7   =  -7>. 


Hence,  the  position  of  the  resultant  is  completely  determined. 


EXAM  PLE 


1.     Required  the  intensity  and  direction  of  the  resultant 
of  three  fo^es  at  right  angles  to  each  pther,  having  the  in- 
tensities 4,  0,  and  6  pounds,  respectively. 
2* 


'64  MECHANICS. 

SOLUTION 

We  have, 
X  =4,  Y=  5,  and  Z  =  6.  .'.     E  =  y/Tl—  8.775. 

Also,  cos  a  =  -4-- ,  cos  p  =  JL  ,  and  cos  y  =  -6f  5  ; 
whence,  n=  62°52'51",  0  =  55°  15 '50",  and  7  =  46°51'43". 

Hence  the  resultant  pressure  is  8.775  lbs.,  and  it  makes,  with 
the  components  taken  in  order,  angles  equal  to  62°  52'  51", 
55°  15'  50",  and  46°  51'  43". 

2.  Three  forces  at  right  angles  are  to  each  other  as  the 
numbers  2,  3,  and  4,  and  their  resultant  is  60  lbs.  What  are 
the  intensities  of  the  forces  ? 

SOLUTION. 

We  have 

Y  =  f X,    Z  =  2X,   and  B  =  60  ; 

Hence, 

60  =  V^rTfTTIP  -   £X<v/29   =   2.6925X 
.-.     X  =   22.284. 
The  components  are,  therefore, 

22.284  lbs.,   33.426  lbs.,   and  44.568  lbs. 
Projection  of  Forces. 

32.  If  planes  be  passed  through  the  extremities  of  a 
force,  perpendicular  to  the  direction  of  any  straight  line, 
that  portion  of  the  line  intercepted  between  them  is  the  pro- 
jection of  the  force  upon  the  line.  The  operation  of  resolv- 
ing forces  into  components  in  the  direction  of  rectangular 
axes, is  nothing  more  than  that  of  finding  their  projections 
upon  these  axes. 

If  two  straight  lines  be  drawn  through  the  extremities  of 
a  force,  perpendicular  to  any  plane,  and  the  points  in  which 
they  me  •!  t!i  ■  plane  be  joined  by  a  straight  line,  this  line  is 
the  projection  of  the  force  upon  tht  phine. 


COMPOSITION    AND    RESOLUTION    OF    FORCES.  35 

If  we  denote  any  force  by  P,  and  the  angle  which  it 
makes  with  any  line  or  plane  by  a,  P  cos  a  will  represent 
the  projection  of  the  force  on  the  line  or  plane.  In  both 
cases  the  projection  of  the  force  is  its  effective  component  in 
the  direction  of  the  line  or  plane  upon  which  it  is  projected. 
Composition  of  a  Group   of  Forces  in  a  Plane. 

33.  Let  P,  P',  P",  <fca,  denote  any  number  of  forces 
lying  in  the  same  plane,  and  applied  at  a  common  point,  and 
represent  the  angles  which  they  make  with  the  axis  of  X  by 
.*,  a',  a",  &c.  Their  components  in  the  direction  of  the  axis 
^f  X  are  P  cos  a,  P'  cos  a',  P"  cos  a",  &c,  and  their  com- 
ponents in  the  direction  of  the  axis  of  Y,  are  P  sin  a, 
P'  sin  a',  P"  sin  ar,  &c. 

If  we  denote  the  resultant  of  the  group  of  components 
which  are  parallel  to  the  axis  of.  X  by  X,  and  the  resultant 
of  the  group  parallel  to  the  axis  of  Y  by  T,  we  shall  have, 
(Art.  26), 

X=  2  (P  cos  a),    and  Y  =  2  (P  sin  a)   .    .    (5.) 

The  resultant  of  JTand  Yis  the  same  as  the  resultant  of  the 
given  forces.  Denoting  this  resultant  by  P,  and  recollecting 
that  A"  and  Y  are  perpendicular  to  each  other,  we  have,  as 
in  Article  30, 

B  =  ■x/xr+rr*     .    .    .    .    ( 6.) 

(f  we  denote  the  angle  which  the  resultant  makes  with  the 
ayis  of  X  by  a,  we  shall  have,  as  in  Article  30, 

X  Y 

cos  a  =  -77  ,    and  sin  a  —  -75- 
lx  It 

EXAMPLES. 

1.  Three  forces,  whose  intensities  are  respectively  equal 
to  50,  40,  and  70,  lie  in  the  same  plane,  and  are  applied  at 
the  same  point,  and  make  with  an  axis  through  that  point, 
angles  equal  to  15°,  30°,  and  45°,  respectively.  Required 
the  intensity  and  direction  of  the  resultant. 


36  MECHANICS. 


S  OLUTION. 

We  have, 

^Y  =  50  cos  15°  +  40  cos  30°  +  70  cos  45°  =  132.435, 
and 

Y  =  50  sin  15°  +  40  sin  30°  +  70  sin  45°  =  32.44  ; 
"whence, 

R  =   -v/6798  +  17539  =   156. 

1 qo ao x 

and  cos  a   =         '    — •  .-.    a  =  31°  54' 12". 

156      ' 

The  resultant  is  156,  and  the  angle  which  it  makes  with  the 
axis  is  equal  to  31°  54'  12". 

2.  Three  forces  4,  5,  and  6,  lie  in  the  same  plane,  making 
equal  angles  with  each  other.  Required  the  intensity  of 
their  resultant  and  the  angle  which  it  makes  with  the  least 
force. 

SOLUTION. 

Take  the  least  force  as  the  axis  of  X.  Then  the  angle 
between  it  and  the  second  force  is  120°,  and  that  between  it 
and  the  third  force  is  240°.     We  have 

X  =  4  +  5  cos  120°  4-  6  cos  240°  =   -  1.5  ; 
Y=  5  sin  120°   +  6   sin  240°  -    —    .866; 

r-  1.5  .866 

■•'     *  =   V3'    C°S   °  =  ~  17732'  *ma  =  "  1^32; 
.-.    a  =  210°. 

3.  Two  forces,  one  of  5  lbs.  and  the  other  of  7  lbs.,  are 
applied  at  the  same  point,  and  make  wit li  each  other  an 
angle  of  12©°.     What  is  the  intensity  of  their  resultant? 

An*.    6.24  lbs. 

Composition  of  a  Group  of  Forces  in  Space. 

34.  Let  the  forces  be  represented  by  1\  P',  P",  &c. 
The  angles  which  they  make  with  the  axis  of  A",  by  ,  ',  a", 
Ac.,  the  angles  which  they  make  with  the  axis  of  }  ,  >\  , 
Q\  3'\  <tc,  and  the  angles  which   they  make  with  the  axis 


COMPOSITION    AND    RESOLUTION    OF   FORCES,  37 

of  Z  by  /,  y\  y'\  &c.  Resolving  each  force  into  compo- 
nents, respectively  parallel  to  the  three  co-ordinate  axis, 
and  denoting  the  resultants  of  the  groups  in  the  directions 
of  the  respective  axes  by  X,  I",  and  Z,  we  shall  have,  as  in 
the  preceding  article, 

X  =   2    (P  COS  a),      T  =   2   (P  COS  /3),    Z  =   2    (P  COS  7.) 

If  we  denote  the  resultant  of  the  system  by  P,  and  the 
angles  which  it  makes  with  the  axes  by  a,  b,  and  c,  we  shall 
have,  as  in  Article  31, 


R  =  y/X1  +  Y*  +  Z\ 

X  f,  Y  A  Z 

cos  a  =  t=5  ,    cos  o  =   -5,   and  cos  c  —  -^  • 

The  application  of  these  formulas  is  entirely  analogous  to 
that  of  the  formulas  in  the  preceding  article. 

Expression  for  the  Resultant  of  two  Forces. 

35.  Let  us  consider  two  forces,  P  and  P',  situated  in 
the  same  plane.  Since  the  position 
of  the  co-ordinate  axes  is  perfectly 
arbitrary,  let  the  axis  of  X  be  so 
taken  as  to  coincide  with  the  force 
?;  a  will  then  be  equal  to  0,  and  we 
shall  have  sin  a  —  0,  and  cos  a  =  1.  Fig.  12. 

The  value   of  X  (Equation  5),  will 

become  P  +  P'  cos  a',  and  the  value  of  Y  will  be- 
come P'  sin  a'.  Squaring  these  values,  substituting 
them  in  Equation  ( 6 ),  and  reducing  by  the  relation 
sm5  a'  -f  cos2  a'  =  1,  we  have, 

R  =  </P"  +  P2  +  2PP'  cos  a'        .     (  V.) 

The  angle  a'  is  the  angle  included  between  the  given 
forces.     Hence, 

The  resultant  of  any  two  forces,  applied  at  the  same 
peint,  is  equal  to  the  square  root  of  the  sum  of  the  squares 


38 


MECHANICS. 


of  the  two  forces,  plus  twice  the  product  of  the  forces  Into 
the  cosine  of  their  included  angle. 

If  we  make  a'  greater  than  90°,  and  less  than  270°,  its 
cosine  will  be  negative,  and  we  shall  have, 


If  we  make  a' 
have, 


B  =:  -v/P2  +  Pn  -  2PPf  cos  a'. 

0,  its  cosine  will  be  1,  and  we  shall 
B  =  P  +  P'. 


If  we  make  a'   =    90°,  its  cosine  will  be  equal  to  0,  and 
we  shall  have, 

B  =  t/P*  +  P'\ 

180°,  its  cosine  will  be  —   1,  and  we 
B  =  P  -  P'. 


If  we  make  a' 
shall  have, 


The  last  three  results  conform  to  principles  already  de- 
duced. Let  P  and  Q  be  two  forces, 
and  B  their  resultant.  The  figure 
QP  being  a  parallelogram,  the 
side  PB  is  equal  to  Q.  From  the 
triangle  OBP  we  have,  in  accor- 
dance with  the  principles  of  trigo- 
nometry, 


Fiir.  13. 


P  :  Q  :  B  : :  sin  OBP  :  sin  BOP  :  sin  OPB.      ( 8.) 


<i~~- 


If  we  apply  a  force  B'  equal  and  directly  opposed  to  B, 
the  forces  P,  §,  and  B',  will  be  in 
equilibrium.  The  angles  OBP, 
and  QOB',  being  opposite  exte- 
rior and  interior  angles,  are  sup- 
plements of  each  other ;  hence, 
sin  OBP  =  sin  QOB'.  The 
angles  BOP,  and  BOB',  are  ad- 
jacent, and,  consequently,  Bupple- 
mentarv;    hence,    sin    BOP    -    sin    POB'.     The   angles 


Fig  14. 


COMPOSITION    AND    RESOLUTION    OF    FORCES. 


3D 


OPR,  and  POQ,  are  interior 
angles  on  the  same  side,  and,  con- 
sequently, supplementary ;  hence, 
sin  OPR  =  sin  POQ.  We  have 
also  R  —  R'.  Making  these  sub- 
stitutions in  the  preceding  propor- 
tion, we  have, 


P  :   Q  :  R' 


Fig  14 

sin  QOR'  :   sin  POP'  :   sin  POQ. 


Hence,  if  three  forces  are  in  equilibrium,  each  is  propor- 
tional to  the  sine  of  the  angle  between  the  other  two. 


EXAMPLES 


1.  Two  forces,  P  and  Q,  are  equal  in  intensity  to  24  and 
30,  respectively,  and  the  angle  between  them  is  105°.  What 
is  the  intensity  of  their  resultant  ? 


JS  =  V242  +  302  +  2   X  24  X  30  cos  105°  =  33.21. 

2.  Two  forces,  P  and  Q,  whose  intensities  are  respec- 
tively equal  to  5  and  12,  have  a  resultant  whose  intensity  is 
13.     Required  the  angle  between  them. 


13  =  y25  -f  144  +  2    X  5   X   12  cos  a. 

.*.     cos  a  =  0,     or  a  =  90°.     Ans. 

3.  A  boat  is  impelled  by  the  current  at  the  rate  of  4 
miles  per  hour,  and  by  the  wind  at  the  rate  of  7  miles  per 
hour.  What  will  be  her  rate  per  hour  when  the  direction 
of  the  wind  makes  an  angle  of  45°  with  that  of  the  current  ? 


R  =  06  +  49  +  2   x  4  x  7  cos  45°  =  10.2m.     Ans. 

4.  A  weight  of  50  lbs.,  suspended  by  «i  string,  is  drawn 
aside  by  a  horizontal  force  until  the  string  makes  an  angle 
of  30°  with  the  vertical.  Required  the  value  of  the  hori- 
zontal force,  and  the  tension  of  the  string;. 

Ans.     28.8675  lbs.,  and  57.735  lbs 


40  MECHANICS. 

5.  Two  forces,  and  their  resultant,  are  all  equal.  What 
is  the  value  of  the  angle  between  the  two  forces  ?  120°. 

6.  A  point  is  kept  at  rest  by  three  forces  of  6,  8,  and  11 
lbs.,  respectively.  Required  the  angles  which  they  make 
with  each  other. 

SOLUTION. 

We  have  P  =  8,  Q  =  6,  and  B'  =  11.  Since  the 
forces  are  in  equilibrium,  we  shall  have  P'  =  P  =11; 
hence  from  the  preceding  article, 


11  =  y64  -f  36  +  90  cos  QOP; 
.-.     cos  QOP  =  H;    or,  QOP  =  77°  21'  52". 

From  the  last  proportion  we  have, 

ainiw     e  sin POii,  _  53224 

sin  §  OP         11' 
or,  FOR'  =  147°  50'  34". 

Also,  S4^£l'  =  n  i      •'•   Sk  «0jR'  =  •'0965 » 

sm  C^tAr  11 

or,  00i2'  =  134°  47' 34" 


Principle  of  Moments. 

36.  The  moment  of  a  force,  with  respect  to  a  point,  is 
the  product  obtained  by  multiplying  the  intensity  of  the 
force  by  the  perpendicular  distance  from  the  point  to  the 
line  of  direction  of  the  force. 

The  fixed  point  is  called  ih  e  centre  of  moments  :  the  per- 
pendicular distance  is  called  tlie  lever  arm  of  the  force)  and 
the  rnomenl  itself  measures  the  tendency  of  the  force  to 
produce  rotation  about  the  centre  of  moments. 


COMPOSITION    AND    RESOLUTION    OF    FORCES. 


41 


Fig.  15. 


Let  P  and   Q  be  any  two 

forces,  and  It  their  resultant ; 
assume  any  point  C,  in  their 
plane,  as  the  centre  of  moments, 
and  from  it,  let  fall  upon  the  di- 
rections of  the  forces,  the  per- 
pendiculars, Op,  Cq,  and  Cr; 
denote  these  perpendiculars  res- 
pectively by  p,  q,  and  r.  Then  will  Pp,  Qq,  and  Br,  be 
the  moments  of  the  forces  P,  Q,  and  B.  Draw  CO,  and 
from  P  let  fall  the  perpendicular  PS,  upon  OP.  Denote 
the  angle  POP,  by  a,  the  angle  BOQ,  or  its  equal,  OBP, 
by  p,  and  the  angle  BOG  by  9. 

Since  PB  =  $,  we  have  from  the  right-angled  triangles 
OPS  and  PBS,  the  equations, 

B  =  Q  cos  fi  4-  JP  cos  a. 
0    =  Q  sin  /3  —  P  sin  a. 

Multiplying  both  members  of  the  first  equation  by  sin  9, 
and  both  members  of  the  second  by  00s  9,  then  adding  the 
resulting  equations,  we  find, 

B  sin  9  =  Q  (sin  9  cos  /3  -f  sin  /3  cos  9)  + 
jP  (sin  9  cos  a  —  sin  a  cos  9). 

Whence,  by  reduction, 

B  sin  9  =  Q  sin  (9  -f  (3)  +  P  sin  (9  —  a). 
From  the  figure,  we  have, 


P 


sin  <P  =  ~777>>    sin  (?  -  a)  =  TTTh    and  sin  fa  +  •/3) 


0(7 


06" 


00 


Substituting  in  the  preceding  equation,  and  reducing,  we 
have, 

Er=Qq  +  Pp. 

When  the  point  C  fills  within  the  angle  POB,  9  —  a 
becomes  negative,  and  the  equation  just  deduced  becomes 


Br=Qq-  Pp. 


42  MECHANICS. 

Hence,  we  conclude  in  all  cases,  that  the  moment  of  the 
resultant  of  two  forces  is  equal  to  the  algebraic  sum  of  the 
moments  of  the  forces  taken  separately. 

If  we  regard  the  force  Q  as  the  resultant  of  two  others, 
and  one  of  these  in  turn,  as  the  resultant  of  two  others,  and 
so  on,  the  principle  may  be  extended  to  any  number  of 
forces  lying  in  the  same  plane,  and  applied  at  the  same 
point.  This  principle  may,  in  the  general  case,  be  expressed 
by  the  equation 

I?r  =  2(Pp) (9.) 

That  is,  the  moment  of  the  resultant  of  any  number  of 
forces,  lying  in  the  same  plane,  and  applied  at  the  same, 
point,  is  equal  to  the  algebraic  sum  of  the  moments  of  the 
forces  taken  separately. 

This  is  called  the  principle  of  moments. 

The  moment  of  the  resultant  is  called  the  resultant  mo- 
ment ;  the  moments  of  the  components  are  called  compo- 
nent moments /  and  the  plane  passing  through  the  resultant 
and  centre  of  moments,  is  WiZ  plane  of  moments. 

When  a  force  tends  to  turn  its  point  of  application  about 
the  centre  of  moments,  in  the  direction  of  the  motion  of 
the  hands  of  a  watch,  its  moment  is  considered  positive ; 
consequently,  when  it  tends  to  produce  rotation  in  a  contrary 
direction,  the  moment  must  be  negative.  If  the  resultant 
moment  is  negative,  the  tendency  oi  the  system  is  to  pro- 
duce rotation  in  a  negative  direction  about  the  centre  of 
moments.  If  the  resultant  moment  is  0,  there  is  no  ten- 
dency to  produce  rotation  in  the  system.  The  resultant 
moment  may  become  0,  either  in  consequence  of  the  lever 
arm  becoming  0,  or  in  consequence  of  the  resultant  itself 
being  equal  to  0.  In  the  former  case,  the  centre  of  mo- 
ments lies  upon  the  direction  of  the  resultant,  and  the  nu- 
merical value  of  the  sum  of  the  moments  of  the  forces 
which  tend  to  produce  rotation  in  one  direction,  is  equal  to 
that  of  those  which  tend  to  produce  motion  in  a  contrary 
direction.  In  the  latter  case,  the  system  of  forces  is  in 
equilibrium. 


COMPOSITION    AND    RESOLUTION    OF    FORCES.  43 

Moments,  with  respect  to  an  Axis. 

37.  To-  form  an  idea  of  the  moment  of  a  force  with 
respect  to  a  straight  line,  taken 

as  an  axis  of  moments.  Let  P 
represent  any  force,  and  let  the 
axis  of  Z  be  assumed  so  as  to 
coincide  with  the  axis  of  mo- 
ments. Draw  the  straight  line 
AB  perpendicular,  both  to  the 
direction  of  the  force  and  to  the 
axis  of  moments  ;    at  the  pomt  Fig.  16. 

A,  in  which   this  perpendicular 

intersects  the  direction  of  the  force,  let  the  force  P  be 
resolved  into  two  components,  P"  and  P' ,  the  first  parallel 
to  the  axis  of  Z,  and  the  second  at  right  angles  to  it.  The 
former  will  have  no  tendency  to  produce  rotation,  the  latter 
will  tend  to  produce  rotation,  which  tendency  will  be  mea- 
sured by  P'  x  AB\  this  product  is  the  moment  of  the 
force  P  with  respect  to  the  axis  of  moments,  and  is  evi- 
dently equal  to  the  moment  of  the  projection  of  the  force 
upon  a  plane  at  right-angles  to  the  axis,  taken  with  respect 
to  the  point  in  which  this  axis  pierces  the  plane  as  a  centre 
of  moments. 

If  there  are  any  number  of  forces  situated  in  any  manner 
in  space,  it  is  clear  from  the  preceding  principles  that  their 
resultant  moment,  with  respect  to  any  straight  line  taken  as 
an  axis  of  moments,  is  equal  to  the  algebraic  sum  of  the 
component  moments  icith  respect  to  the  same  axis. 
Principle  of  Virtual  Moments. 

38.  Let  P  represent  a  force  applied   to   the  material 
point  0 ;  let  the  point  0  be  moved  by 

an    extraneous  force    to    some    position,      p'Op jp 

O,  very  near    to    0 ;    project   the  path       £''    *'g 

OC    upon   the  direction    of  the    force;  Fig.  17. 

the  projection    Op,  or  Op\  is  called  the 

virtual  velocity  of  the  force,  and  is  taken  positively  when 

it  falls  upon  the  direction  of  the  force,  as  Op,  and  nega- 


44  MECHANICS. 

tively  when  it  falls  upon  the  prolongation  of  the  force,  as 
Op'.  The  product  obtained  by  multiplying  any  force  by  its 
virtual  velocity  is  called  the  virtual  moment  of  the  force 

Assume  the  figure  and  nota- 
tion of  Article  36.  Op,  Oq,  and 
Or  are  the  virtual  velocities  of  _  Jf"    ~~s 

the  forces  P,  §,  and  R.  Let 
us  denote  the  virtual  velocity  of  o°^ 
any  force  by  the  symbol  of  va- 
riation 8,  followed  by  a  small 
letter  of  the  same  name  as  that 
which  designates  the  force. 

We  have  from  the  figure,  as  in  Article  36,  the  relations, 

R  =  P  cos  a.  +  Q  cos  (3. 

0   =  P  sin  a  —  Q  sin  (3. 

Multiplying  both  members  of  the  first  by  cos  9,  and  of  the 
second  by  sin  9,  and  adding  the  resultant  equations,  we  have, 

R  cos  9  =  P  (cos  a  cos  9  4-  sin  a  sin  9)  4 
Q  (cos  9  cos  (3  —  sin  9  sin  (3). 

Or,  by  reduction, 

^  R  cos  9  =  P  cos  (9  —  a)  4-  Q  cos  (9  4-  (3). 

But,  from  the  right-angled  triangles  COp,  COq,  and  COr, 
we  have, 

cos 9  =  -0^,  cos  (9  —  a)  =  —-,  and  cos  (9  +  P)  -   0^\ 

Substituting  these  in  the  preceding  equation,  and  reducing, 
we  have, 

JRSr  =  JPSp  +   Q8q. 

Hence,  the  virtual  moment  of  the  resultant  of  two  forces, 
is  equal  to  the  algebraic  sum  of  the  virtual  moments  of  t/ie 
two  forces  taken  separately. 


COMPOSITION    AND    RESOLUTION    OF    FORCES.  45 

If  we  regard  the  force  Q  as  the  resultant  of  two  other 
forces,  and  one  of  these  as  the  resultant  of  two  others,  and 
so  on,  the  principle  maybe  extended  to  any  number  offerees, 
applied  at  the  same  point.  This  principle  may  be  expressed 
by  the  following  equation : 

JRor  =  2  (JPSp)      ....     (10.) 

Hence,  the  virtual  moment  of  the  resultant  of  any  num- 
ber of  forces  applied  at  the  same  point,  is  equal  to  the  alge- 
braic sum  of  the  virtual  moments  of  the  forces  taken  sepa- 
rately. 

This  is  called  the  principle  of  virtual  moments.  If  the 
resultant  is  equal  to  0,  the  system  is  in  equilibrium,  and  the 
algebraic  sum  of  the  virtual  moments  is  equal  to  0  ;  con- 
versely, if  the  algebraic  sum  of  the  virtual  moments  of  the 
forces  is  equal  to  0,  the  resultant  is  also  equal  to  0,  and  the 
forces  are  in  equilibrium. 

This  principle,  and  the  preceding  one,  are  much  used  in 
discussing  the  subject  of  machines. 

Resultant  of  parallel  Forces. 
39.     Let  P  and  Q  be  two  forces  lying  in  the  same  plane, 
and  applied  at  points  invariably 
connected,  for    example,    at    the 
points  31  and  JST  of  a  solid  body.  -^ v 

Their  lines  of  direction  being  pro-  ^^\— ^9. 

longed,  will  meet  at  some  point        ftor<^Li_    \  p 

0  ;  and  if  we  suppose  the  points  ^t 

of  application  to  be  transferred  to  Fig-  i» 

0,  their  resultant  may  be  deter- 
mined by  the  parallelogram  of  forces.  The  direction  of  the 
resultant  will  pass  through  O.  (Art.  2V.)  Whether  the 
forces  be  transferred  to  0  or  not,  the  direction  of  the  resul- 
tant will  always  pass  through  O,  and  this  whatever  may  be 
the  value  of  the  included  angle.  Xow,  supposing  the  points 
of  application  to  be  at  M  and  jV,  let  the  force  Q  be  turned 
about  JV  as  an  axis.     As  it  approaches  parallelism  with  P, 


46  MECHANICS. 

the  point  0  will  recede  from  Maud  iVJ  and  the  resultant  will 
also  approach  parallelism  with  P. 
Finally,  when  Q  becomes  parallel 
to  P,  the  point  0  will  be  at  an 
infinite  distance  from  M  and  JV, 
and  the  resultant  will  also  be  par-  ,,^' 

allel  to  P  and  Q.     In  any  position       °^~ 
of  P  and  Qi  the  value  of  the  re- 
sultant,   denoted   by  H,    will  be 
given  by  the  equation  (Art.  36), 

P  —  Pcosa  -f  Qcosfi. 

When  the  forces  are  parallel,  and  lying  in  the  same  direc- 
tion, we  shall  have  a  —  0,  and  /3  =  0;  or,  cos  a  =  1,  and 
cos  [3  =  1.     Hence, 

It  =  P  +  Q. 

If  the  forces  lie  in  opposite  directions,  we  shall  have 
a    —    0,     and    /3    =    180°;    or, 
cos  a  =    1,    and  cos  (3    —    —  1. 

Hence,  s''  ^N, 

P  =  P-  Q. 


That  is,    ?Ae  resultant  of  ttco       qU. -^ 

parallel  forces  is  equal  in  inten- 


Qr-l L V- 


«7y  to  £/*e  algebraic  sum  of  the  M 

forces,  and  its  line  of  direction  Fis 20- 

is  par  allel  to  that  of  the  two  forces. 

If  we  regard  Q  as  the  resultant  of  two  parallel  forces,  and 
one  of  these  as  the  resultant  of  two  others,  and  so  on,  the 
principle  may  be  extended  to  any  number  of  parallel  forces. 
Denoting  the  resultant  of  a  group  of  parallel  forces, 
J\  P\  P",  &c,  by  P,  we  have, 

*  =  2(P) (11.; 

That  is,  the  resultant  of  a  group  of  parallel  forces  is 
equal  in  intensity  to  the  algebraic  sum  of  the  forces.  Ps 
line  of  direction  is  also  par 'allel  to  that  of  the  given  forces. 


No_ 

^Q 

9/     'C 

/  .  ...   i 

-^2 

L 


COMPOSITION    AND    RESOLUTION    OF   FORCES.  47 

Point  of  Application  of  the  Resultant. 
40.     Let  P  and  Q  be  two  parallel  forces,  and  R  their 
resultant.     Let  M  and  JV  be 
the  points  of  application  of 
the   two   forces,    and   S  the 
point  in  which  the  direction 
of   P    cuts    the    line    MN.       K^ 
Through  JV  draw  JSTZ  per-  Fi   2 

pendicular  to  the  general  di- 
rection of  the  forces,  and  assume  the  point  (7,  in  which  it 
intersects  the   line  of  direc- 
tion of  P.  as  a  centre  of  ino-  ST 

.  ft-* 7? 

nients.     Since  the  centre  of  /  ; 

...  „  -My ! >-i? 

moments  is   on   the  line  of  /      [L 

direction    of   the    resultant,  Sol i — >_j», 

the  lever  arm  of  the  resultant 

will  be  0,  and  we  shall  have, 

from  the  principle  of  moments  (Art.  36), 


C 

Fig. 


Px  CZ=  Q  x  CM, 
or,  P  :  Q  :  :  CJV  :  CZ. 

But,  from  the  similar  triangles  CJSTS  and  ZNM,  we  have, 

CJST:  CZ  :  :  8HT  :  SM. 
Combining  the  two  proportions,  we  have, 
P  :  Q  :  :  SJST  :  SM. 

That  is,  the  line  of  direction  of  the  resultant  divides  the 
line  joining  the  points  of  application  of  the  components, 
inversely  as  the  components. 

From  the  last  proportion,  we  have,  by  composition, 

P  :  Q  :  P+  Q  :  :  SIT:  8M:  SJV  +  SM; 
and,  by  division, 

P  :  Q  :  P  -  Q  :  :  SJST  •  SM  :  S2T-  SM. 


48 


MECHANICS. 


When  the  forces  act  in  the  same  direction,  P  +  Q  will 
be  their  resultant,  and  SN  -f  SM  will  equal  MN.  Since 
P  +  Q  is  greater  than  either  P  or  Q,  MN  will  be  greater 
than  either  SN  or  SM,  which  shows  that  the  resultant  lies 
between  the  components. 

When  the  forces  act  in  contrary  directions,  P  —  Q  will 
be  their  resultant,  and  SN  —  SM  will  equal  MN  Since 
P  —  Q  is  less  than  P  (supposed  the  greater  of  the  compo- 
nents), MN  will  be  less  than  SN,  which  shows  that  the 
resultant  lies  without  both  components,  and  on  the  side  of 
the  greater. 

Substituting  in  the  preceding  proportions,  for  P  -f-  §, 
P-  Q,  SN  +  SM,  and  SN  —  S3I,  their  values,  we  have, 

P  :    (2  :  E  :  :  ^  :  SM  :  JOE  .  .  .  (8)'. 

That  is,  of  two  parallel  forces  and  their  resultant,  each  is 
proportional  to  the  distance  between  the  other  two. 


H 


« 


31 

Fig.  23. 


-Si" 


Geometrical  Composition  and  Resolution  of  Parallel    Forces. 

41.     The  preceding  principles  give  rise  to  the  following 
geometrical  constructions :  /jp* 

1.  To  find  the  resultant  of  two 
parallel  forces  lying  in  the  same  direc- 
tion : 

Let  P  and  Q  be  the  forces,  M  and 
N  their  points  of  application.      Make 

MQ'  =  Ci  *n<1  #2*  =  7>;  draw  P'C'i 

cutting  JAV  in  S ;  through  £  draw  SP 
parallel  to  MP,  and  make  it  equal  to 
P  +  Q:  it  will  be  the  resultant. 

For,  from  the  similar  triangles  P'SN 
and  Q'SM,  we  have, 


PN  :    Q'M  :  :  SN  :  SM;    or,  P  :    #  :  :  £iV  :  £JSf. 

After  the  construction  is  made,  the  distances  MS  and 
NS  may  be  measured  by  a  scale  of  equal  parts. 


COMPOSITION    AND    RESOLUTION    OF    FORCES. 


49 


EXAMPLE. 

Given  P  =  9  lbs,    Q  =  6  lbs.,    and  JOT  =  30  in.     Re- 
quired MS. 

We  have  P  =  15,  hence, 

15  :  6  :  :  30  :  J/# ;         .-.     Jf/S  =  12  in.     -4ws. 

2.  To  find  the  resultant  of  two  parallel  forces  acting  in 
opposite  directions: 

Let  P  and  Q  be  the  forces,  M  and 
N  their  points  of  application.  Prolong 
QNxS&NA  =  P,  and  make  MB  =  Q; 
draw  AP,  and  produce  it  till  it  cuts 
NM  produced  in  S ;  draw  SP  parallel 
to  MP,  and  make  it  equal  to  BP,  it 
will  be  the  resultant  required. 

For  from  the  similar  triangles  SNA 
and  SMP,  we  have, 

AN  :  P3I :  :  SN  :  S3I;     or,    P  :    Q 

EXAMPLE. 

Given    P  =   20  lbs.,     Q   =  8  lbs.,    and  _#3f  =   18  in. 
Required  #2^ 

We  have  i?  =  20  —  8  =  12;  hence,  from  Proportion  (8)! 

12  :  20  :  :  18  :  SN:         .'.     SN  =  30  in.     Ans. 


/B 


3fr- 


T—& 


3.  To  resolve  a  given  force  into  two  parallel  components 
lying  in  the  same  direction,  and  applied  at  given  points  : 

Let  P  be  the  given  force,  M  and 
N  the  given  points  of  application. 
Through  M  and  N  draw  lines  parallel 
to  P.  Make  MA  =  P,  and  draw  AN, 
cutting  P  in  P  ;  make  MP  —  SP  and 
NQ  =  i?/?;  they  will  be  the  required 
components. 
3 


n 


/ 


Fig.  2.-V 


50  MECHANICS. 

For,  from  the  similar  triangles  AMX  and  BSXy 


■ 7" 


B 


;q 


Fig.  25. 


Re- 


J?£:  AM:  :  SIT:  MX ; 
or,       J5S  :     i2     :  :  £^T :  J/^ 

But,  from  Proportion  (8)', we  have, 

P  :  B  :  :  SA"  :  3IX; 

.-.     .££  ^  P,  and  BR  =  §. 

EXAMPLE. 

Given  i2  =  24  lbs.,  S3I  =  V  in.,  and  SX  =  5  in. 
quired  P  and  §. 

From  Proportion  (8),  we  have, 

12  :  7  :  :  24  :    Q  ;         .-.     g  =  14  lbs. 
12  :  5  :  :  24  :  P  ;         .-.     P  =  10  lbs. 

4.  To  resolve  a  given  force  into  parallel  components  lying 
in  opposite  directions,  and  applied  at  given  points.  Both 
points  of  application  must  lie  on  the  same  side  of  the  given 
force.  Let  B  bo.  the  given  force,  M  and  X  the  given 
points  of  application.  Through  M 
and  A7"  draw  lines  parallel  to  B  ;  make 
XB  —  B,  and  draw  B3I;  through 
S,  draw  >S'J.  parallel  to  MB;  then 
will  XA  and  BA  be  equal  to  the  in- 
tensities of  the  components.  Make 
MP  =  AN,  and  XQ  =  AB,  and 
they  will  be  the  components.  For, 
from  the  triangles  ASX,  and  BMX, 
we  have, 


^""Ifr 


]    g.26 


AX  :  BN  :  :  SX:  3fX;  or,  AX  :  B  :  :  SX  :  J/iV7: 
But,  from  Proportion  (8)',  we  have, 
P  :  B  :  :  SX:  3IX;         /.     AX  =  P,  and  AB  =  Q. 


COMPOSITION    AND    RESOLUTION    OF    FORCES. 


51 


r 


4<1 


F 


p- 


EX  AMPLE. 

Given  R  =  24  lbs.,  SJST  =  18  in.,  and  SM  =  9  in.  Re- 
quired P  and  §. 

From  Proportion  (8)',  we  have, 

P  :  24  :  :  18  :  9 ;         .\     P  =  48  lbs. 
§  :  24  :  :     9:9;         /.     §  =  24  lbs. 

R  =  P  -  Q  =  24  lbs. 

5.  To  find  the  resultant  of  any  number  of  parallel  forces. 

Let  P,  P',  P",  P'",  be  such  a  system  of  forces.  Find 
the  resultant  of  P  and  P\  by  the  rule 
already  given,  it  will  be  R'  =  P  +  P' ; 
find  the  resultant  of  P'  and  P', 
it  will  be  R"  =  P  +  P  +  P" ;  find 
the  resultant  of  P"  and  P'",  it  will  be 
R  =  P  +  P  +  P"  +  P".  If  there 
is  a  greater  number  of  forces,  the 
operation  of  composition  may  be  con- 
tinued ;  the  final  result  will  be  the  re- 
sultant of  the  system.  If  some  of  the 
forces  act  in  contrary  directions,  combine  all  which  act  in 
one  direction,  as  just  explained,  and  call  their  resultant  R'; 
then  combine  all  those  which  act  in  a  contrary  direction, 
ind  call  their  resultant  R"  ;  finally,  combine  R'  and  R"  by 
a  preceding  rule  ;  their  resultant  R  will  be  the  resultant  of 
the  system. 

If  R'  —  R",  the  resultant  will  be  0,  and  its  point  of  ap- 
plication will  be  at  an  infinite  distance.  In  this  case,  the 
forces  reduce  to  a  couple,  the  efiect  of  which  is  simply  to  pro- 
duce rotation. 

Lever  Arm  of  the  Resultant. 

42.  Let  P,  P',  P",  <fec,  denote  any  number  of  parallel 
forces,  and  p,  p',p",  &c,  their  lever  arms  with  respect  to  an 
axis  of  moments,  taken  perpendicular  to  the  common  direc- 
tion of  the  forces  :  denote  the  lever  arm  of  the  resultant  of 


Fig.  27. 


52  MECHANICS. 

the  system,  taken  with  respect  to  the  same  axis,  by  r      From 
the  principle  of  moments  (Art.  37), 

(P  +  P  +  P"  +  Ac.)*  =  P/?  +  P>'  +  &c. ; 

Hence,  £Ae  lever  arm  of  the  resultant  of  a  system  of  par- 
allel forces,  with  respect  to  an  axis  at  right-angles  to  their 
direction,  is  equal  to  the  algebraic  sum  of  the  moments  of 
the  forces  divided  by  the  algebraic  sum  of  the  forces. 

Centre  of  Parallel  Forces. 

43.  Let  there  be  any  number  of  forces,  P,  P,  P',  <fcc, 
applied  at  points  invariably  connected  together,  and  whose 
coordinates  are  x,y,z\  x\  y\  z  ;  as",  y'\  z"  ;  &c.  Let  P 
denote  their  resultant,  and  represent  the  co-ordinates  of  its 
point  of  application,  by  x^  y1?  and  zl ;  denote  the  angles  made 
by  the  common  direction  of  the  forces  with  the  axes  of 
X,  T,  and  Z,   by  a,  ft  and  /. 

Suppose  each  force  resolved  into  three  components,  re- 
spectively parallel  to  the  co-ordinate  axes,  the  points  of 
application  being  unchanged : 

The  components  parallel  to  the  axis  of  AT  are, 

Pcosa,    Pcosa,    P'cosa,  &c.j    Pcosa  ; 

those  parallel  to  the  axis  of  I^are, 

Pcos/3,    Pcos  ft   P"cosft  <fcc.,   Pcos/3 ; 

ind  those  parallel  to  the  axis  of  Z  are, 

Pcos/,    Pcos/,   P'cos/,  &c,   Pcos/. 

If  we  take  the  moments  of  the  components  parallel  to  the 
txis  of  Z,  with  respect  to  the  axis  of  Y,  as  an  axis  of  mo- 
ments, we  shall  have,  for  the  lever  arms  of  the  components, 
r,  x\  x",  &c. ;  and  from  the  principle  of  moments  (Art.  37), 

/?cos/  xx  —  Pcos/  x  +  Pcos/  x'  +  <^c' 


COMPOSITION    AND    RESOLUTION    OF    FORCES.  53 

Striking  out  the  common  factor  cos  y,  and  substituting 
for  H  its  value,  we  have, 

X(Jfe  =  2(A}$ 

2(F)' 


mce,  x,  = 


In  like  manner,  if  we  take  the  moments  of  the  same  com- 
ponents, with  respect  to  the  axis  of  JT,  we  shall  have, 

yi  -    2(P) 

And,  if  we  take  the  moments  of  the  components  parallel 
to  the  axis  of  Y,  with  respect  to  the  axis  of  X,  we  shall 
have, 

2(P2) 


2,  = 


2(P)' 


Hence  we  have  for  the  co-ordinates  of  the  point  of  appli- 
cation of  the  resultant, 

x  -  W   „  _  JiSfl  andz  _  W  (13) 

These  co-ordinates  are  entirely  independent  of  the  direc- 
tion of  the  parallel  forces,  and  will  remain  the  same  so  long 
as  their  intensities  and  points  of  application  remain  un- 
changed. 

The  point  whose  co-ordinates  we  have  just  found,  is  called 
the  centre  of  parallel  forces. 

Resultant  of  a  Group  of  Forces  in  a  Plane,  and  applied  at  points 
invariably  connected. 

44.  Let  P,  P\  P",  &c,  be  any  number  of  forces  lying 
in  the  same  plane,  and  applied  at  points  invariably  connected 
together ;  that  is,  at  points  of  the  same  solid  body. 


54 


MECHANICS. 


PcosjS 


"^HPcoso. 


Through  any  poiut  0  in  the  plane  of  the  forces,  draw  any 
two  straight  lines,  6LY and  OY, 
at  right  angles  to  each  other,  and 
lying  in  the  plane  of  the  forces  ; 
assume  these  as  co-ordinate  axes. 
Denote    the   angles    which    the 
forces  P,  P',  P'\  &c,  make  with      — 
the  axis    OX,  by  a,  a',  a",  <fce., 
and  the  angles  which  they  make 
with  the  axis  01",  by  /3,  /3',  0",  &c. ;  denote,  also,  the  co- 
ordinates of  the  points  of  application  of  the  forces,  by  #,  y ; 


J 


Fig.  28. 


*>y  ;  «  ,y 


<fcc. 


Let  each  force  be  resolved  into  components  parallel  to  the 
co-ordinate  axes  ;  we  shall  have  for  the  group  parallel  to  the 
axis  of  A", 

Pcos-/,   Pcosa',  P"cosx",  &c. ; 

and,  for  the  group  parallel  to  the  axis  of  Y, 

Pcos.3,   P'cos,-3',   P"cos3",  &c. ; 

The  resultant  of  the  first  group  is  equal  to  the  algebraic 
sum  of  the  components  (Art.  39) ;  denoting  this  by  A"  we 
shall  have, 

X=2(i*50Sa)       .     .     .     .      (14.) 

In  like  manner,  denoting  the  resultant  of  the  second  group 
by   rj  we  shall  have, 


T  =  2(7*508:3) 


(15.) 


The  forces  X  and  Y  intersect  in  a  point,  which  is  the 
point  of  application  of  the  system  of  forces.  Denoting  the 
resultant  by  72,  we  shall  have  (Art.  33), 


n  =  y/  xa  +  y*. 

To  find  the  point  of  application  of  7?,  let  0  be  taken  as  a 
centre  of  moments,  and  denote  the  lever  arms  of  X  and  I" 


COMPOSITION    A  NO    RESOLUTION    OF    FORCES.  55 

by  y}  and  xv  respectively.     From  the  principle  of  Article 
42,  we  shall  have, 

_  2(7*08,3  s) 
^  ~    2(Pcos,3)    •     •    •     •     1  lb') 

2(Peosay) 

^  =  -2cH=?-   •   •   •   <"•> 

II  we  denote  the  angles  which  the  resultant  makes  with 
the  axes  of  AT  and  Y  by  a  and  &  respectively,  we  shall 
have,  as  in  Article  33, 

AT  Y 

cosa  =  ^,    cos  b  —  -=  .     .     .      (18.) 

Equations  (16)  and  (17)  make  known  the  point  of  applica- 
tion, and  Equations  (18)  make  known  its  direction  ;  hence, 
the  resultant  is  completely  determined. 

To  find  the  moment  of  P,  with  respect  to  O  as  a  centre 
of  moments,  let  us  denote  its  lever  arm  by  r,  and  the  lever 
arms  of  P,  P',  P",  &c,  with  respect  to  0,  byjfl,  p\  p",  &c. 

The  moment  of  the  force  Pcosa,  is  Pcosa  y,  and  that 
of  the  force  PcosS,  is  —  Pcosfix.  The  negative  sign  is 
given  to  the  last  result,  because  the  forces  Pcos->.  and 
Pcos/3  tend  to  turn  the  system  in  contrary  directions. 

From  the  principle  of  moments  (Art.  36),  the  moment  of 
P  is  equal  to  the  algebraic  sum  of  the  moments  of  its  com- 
ponents.    Hence, 

Pp  —  Pcosa  y  —  PcosQ  x. 

In  like  manner,  the  moments  of  the  other  component 
forces  may  be  found.  Because  the  moment  of  the  resultant 
is  equal  to  the  algebraic  sum  of  the  moments  of  all  its  com- 
ponents (Art.  36),  we  have, 

Br  =  2(Pp)  =  2  (Pcosa  y  -  Pcos.3  x)      .     (19.) 


r>rPcosa 


56  MECHANICS. 

Resultant  of  a  Group  of  Forces  situated  in  Space,  and  applied  at 
points   invariably  connected. 

45.     Let  P,  P\   P",  <fcc,  be    any   number   of   forces 
situated  in  any  manner  in  space, 
and   applied   at   points    of   the  u^ 

same  solid  body.     Assume  any  S 

point  0  in  space,  and  through 
it  draw  any  three  lines  perpen- 
dicular to  each  other.  Assume 
these  lines  as  axes.  Denote  the 
angles  which  the  forces  P,  jp, 
P",  &c,  make  with  the  axis  of 
X,  by  a,  a',  a'',  <fcc. ;  the  angles  Fig  ^ 

which  they  make  with  the  axis 

of  I",  by  3,  /3',  /3",  &c.  ;  the  angles  which  they  make  with 
the  axis  of  Z,  by  7,  7',  7",  &c,  and  denote  the  co-ordi- 
nates of  their  points  of  application  by  xy  y,  z ;  as',  y',  z  ; 
a",  2/",  ■";  &c. 

Let  each  force  be  resolved  into  components  respectively 
parallel  to  the  co-ordinate  axes. 

We  shall  have  for  the  group  parallel  to  the  axis  of  JT, 

Pcosa,   Pcosa',   P'cosx",  &c. ; 
for  the  group  parallel  to  the  axis  of  3^ 

Pcos/3,   PgobP',   P'cos.3",  &c.  ; 
and  for  the  group  parallel  to  the  axis  of  Z, 

Pcos/,    _Pcosy\    P'cos/",  <fcc. 

Denoting  the  resultants  of  these  several  groups  by  JT,  7] 
and  Z,  we  shall  have, 

X=2(Pco^)  F=2(Pcosft)  andZ  =  2(Pcos7)    .    (20.) 

If  these  three  forces  intersect  at  a  point,  this   point  is 
the  point  of  application  of  the  resultant  of  the  entire  svs« 


COMPOSITION    AND    RESOLUTION    OF    FORCES.  57 

tern.     Denote  this  resultant  by  H ;  then,  since  the  forces 
JT,  Y,  and  Z  are  perpendicular  to  each  other,  we  shall  have, 


n  =  vx*  +  y*  +  z*.   .   .    ( 21.) 

To  find  the  Co-ordinates  of  the  point  of  application  of  R. 

Consider  each  of  the  forces,  -3T,  !PJ  and  Z,  with  respect  to 
the  axis  whose  name  comes  next  in  order,  and  denote  the 
lever  arm  of  -3T,  with  respect  to  the  axis  of  Y,  by  zx ;  that 
of  Y,  with  respect  to  the  axis  of  Z,  by  xx ;  and  that  of  Z^ 
with  respect  to  the  axis  of  JC,  by  yv  We  shall  have  as  in 
the  last  article, 

2(Pcosj8a;)  ' 
Xl  "     J3(JPoos/3) 


Vi  = 


«i  = 


2(Pcos/y) 
2(Pcos/) 

2(Pcosa  g) 
2(Pcosa)     - 


(22.) 


in  which  a^,  yls  and  zD  are  the  co-ordinates  of  the  point  of 
application  of  R. 

Denoting  the  angles  wThich  It  makes  with  the  axes  by 
a,  £,  and  c,  respectively,  we  have,  as  in  the  preceding 
article, 

x      *    y  z     /.ox 

cos  a  =  -r=,    cos  6  =  -=,    cos  c  =  -^  .    .  (23.) 

Jt&  1£  Jf> 

The  values  of  JT,  I7^  and  Z,  may  be  computed  by  means  of 
Equations  (20),  and  these  being  substituted  in  (21),  make 
known  the  value  of  the  resultant.  The  co-ordinates  of 
its  point  of  application  result  from  Equations  (22),  and  its 
line  of  direction  is  shown  by  Equations  (23).  The  intensity, 
direction,  and  point  of  application  being  known,  the  resul- 
tant is  completely  determined. 
3* 


58 


MECHANICS. 


Measure  of  the  tendency  to  Rotation  about  the  Axes. 
46.     Let  A",   Y,  and  Z  denote  the  components  of  the 
resultant   of  the  system,  as  in 
the  last  article,  and  denote,  as  « 

before,  the  co-ordinates  of  the 
point  of  application  of  the  re- 
sultant by  «!,  y1?  and  zv  To  find 
the  resultant  moment,  with  re- 
spect to  the  axis  of  Z,  it  may 

be  observed  that  the  component     -%/—■ yY 

Z,  can  produce  no  rotary  effect,     y  x 

since  it  is  parallel  to  the  axis  of  Fig.  30. 

Z ;  the  moment  of  the  compo- 
nent Y,  with  respect  to  the  axis  of  Z,  is  Yxx ;  the  moment 
of  the  component  A",  with  respect  to  the  same  axis,  is 
— JEy15  the  negative  sign  being  taken  because  the  force  AT 
tends  to  produce  rotation  in  a  negative  direction.  Hence, 
the  resultant  moment  of  the  system,  with  respect  to  the 
axis  of  Z,  is, 

Yxx-  XVl- 

or,  substituting  for  AT  and  Y  their  values,  we  have, 

Yx1  -  Xyl  =  2{T>cos3x-  Pcosay)    .    (24.) 

In  like  manner  for  the  resultant  moment  of  the  system, 
with  respect  to  the  axis  X, 

Zyl  —  Is,  —  2(Pcos7?/  —  Pcos/3  z)    .    ( 25.) 

And  for  the  resultant  moment,  with  respect  to  the  axis 

of  r, 

JTz,  —  Zx1  =  2(Pcosas  —  Pcosyx)      .      (26.) 


Equilibrium  of  Forces  in  a  Plane. 

4  7.     Tn  order  that   a  system  of  forces  lying  in  the  same 

plane,  and   applied   at  points   of  a  free   solid,  may  be   in 

equilibrium,   two   conditions    must  be  fulfilled:    First,  the 

resultant  of  the  system  must  have  no  tendency  to  produce 


EQUILIBRIUM    OF    FORCES.  59 

motion  of  translation ;  and,  secondly,  it  must  have  no 
tendency  to  produce  motion  of  rotation.  Conversely,  if 
these  conditions  are  satisfied,  the  system  will  be  in  equi- 
librium. 

The  first  condition  will  be  fulfilled,  and  will  only  be  ful- 
filled, when  the  resultant  is  equal  to  0  ;  but  from  Art.  44,  we 
have, 

e  =  v'x2  +  3F. 

The  value  of  B  can  only  be  equal  to  0  when  X  —  0,  and 
Y  =  0 ;  or,  what  is  the  same  thing, 

2(Pcos«)  =  0,  and  2(Pcos/3)  =  0     .      (27.) 

The  second  condition  will  be  fulfilled,  and  will  only  be 
fulfilled,  when  the  moment  of  the  resultant,  with  respect  to 
any  point  of  the  plane,  is  equal  to  0,  whence, 

Br  =  0;    or,  Z(Pp)  =  0   .     .     .     (28.; 

Hence,  from  Equations  (27)  and  (28),  in  order  that  a 
system  of  forces,  lying  in  the  same  plane,  and  applied  at 
points  of  a  free  solid  body,  may  be  in  equilibrium,  we  must 
have, 

1st.  The  algebraic  sum  of  the  components  of  the  forces  in 
the  direction  of  any  two  rectangular  axes  separately  equal 
to  0. 

2d.  The  algebraic  sum  of  the  moments  of  the  forces,  with 
respect  to  any  point  in  the  plane,  equal  to  0. 

Equilibrium  of  Forces  in  Space. 
48.  In  order  that  a  system  of  forces  situated  in  any  man- 
ner in  space,  and  applied  at  points  of  a  free  solid  body,  may 
be  in  equilibrium,  two  conditions  must  be  fulfilled.  First,  the 
forces  must  have  no  tendency  to  produce  motion  of  transla- 
tion ;  and  secondly,  they  must  have  no  tendency  to  produce 
motion  of  rotation  about  either  of  the  three  rectangular 
axes.  Conversely,  when  these  conditions  are  fulfilled,  the 
system  will  be  in  equilibrium.     The  first  condition  will  be 


60  MECHANICS. 

fulfilled,  and  will  only  be  fulfilled,  when  the  resultant  is 
equal  to  0.     But,  from  Equation  (21), 


That  this  value  of  it  may  be  0,  we  must  have,  separately, 

X  =  0,    Y=  0,  and  Z  =  0  ; 

or,  what  is  the  same  thing, 

2(Pcosa)  =0,    2(Pcos.3)  =  0,   and2(Pcosy)  =  0   .    (29.) 

The  second  condition  will  be  fulfilled,  and  will  only  be 
fulfilled,  when  the  moments,  with  respect  to  each  of  the 
three  axes,  are  separately  equal  to  0.     This  gives  (Art.  46), 

2(Pcos,o  x  —  Pcosa  y)  =  0  " 

2(Pcos/y  -  Pcos3z)  =  0  I  •     •    (30.) 

2(Pcosa  z  —  Pcos/  x)  =  0  J 

Hence  (Equations  29  and  30),  in  order  that  a  system  ot 
forces  in  space  applied  at  points  of  a  free  solid  may  be  in 
equilibrium : 

1st.  TJie  algebraic  sum,  of  the  components  of  the  forces  in 
the  direction  of  any  three  rectangular  axes  must  be  separate- 
ly equal  to  0. 

2d.  TJie  algebraic  sum  of  the  moments  of  the  forces,  with 
respect  to  any  three  rectangular  axes,  must  be  separately 
equal  to  0. 

Equilibrium  of  Forces  applied  to  a  Revolving  Body. 

49.  If  a  body  is  restrained  by  a  fixed  axis,  about  which 
it  is  free  to  revolve,  we  may  take  this  line  as  the  axis  of  AT. 
Since  the  axis  is  fixed,  there  can  be  no  motion  of  transla- 
tion, neither  ran  there  be  any  rotation  about  either  of  the 
Other  two  axes  of  co-ordinates.  All  of  Equations  (29),  and 
the  first  and  third  of  Equations  (30),  will  be  satisfied  bj 
virtue  of  the  connection  of  the  body  with  the  fixed  axi* 


EQUILIBRIUM    OF    FORCES.  61 

The  second  of  Equations  (30)  is,  therefore,  the  only  one  that 
must  be  satisfied  by  the  relation  between  the  forces.  We 
must  have,  therefore, 

l(Fcosyy  —  Pcos,3  z)  =  0      .     .     (31.) 

That  is,  if  a  body  is  restrained  by  a  fixed  axis,  the  forces 
applied  to  it  will  be  in  equilibrium  when  the  algebraic  sum 
of  the  moments  of  the  forces  with  respect  to  this  axis  is 
§qual  to  0. 


62  MECHANICS. 


CHAPTER  III. 

CENTRE    OF    GRAVITY    AXI)    STABILITY. 

Weight. 

50.  That  force  by  virtue  of  which  a  body,  when  aban- 
doned to  itself,  falls  towards  the  earth,  is  called  the  force 
of  gravity.  The  force  of  gravity  acts  upon  every  particle 
of  a  body,  and,  if  resisted,  gives  rise  to  a  pressure;  this 
pressure  is  called  the  weight  of  the  particle.  The  resultant 
weight  of  all  the  particles  of  a  body  is  called  the  weight  of 
the  body.  The  weights  of  the  particles  are  sensibly  directed 
towards  the  centre  of  the  earth ;  but  this  point  being  nearly 
4,000  miles  from  the  surface,  we  may,  for  all  practical  pur- 
poses, regard  these  weights  as  parallel  forces ;  hence,  the 
weight  of  a  body  acts  in  the  same  direction  as  the  weights 
of  its  elementary  particles,  and  is  equal  to  their  sum. 

Centre  of  Gravity. 

51.  The  centre  of  gravity  of  a  body  is  the  point  of  ap- 
plication of  its  weight.  The  weight  being  the  resultant  of 
a  system  of  parallel  forces,  the  centre  of  gravity  is  a  centre 
of  parallel  forces,  and  so  long  as  the  relative  position  of  the 
particles  remains  unchanged,  this  point  will  retain  a  fixed 
position  in  the  body,  and  this  independently  of  any  parti- 
cular position  of  the  body  (Art.  43).  The  position  of  the 
centre  of  gravity  is  entirely  independent  of  the  value  of  the 
force  of  gravity,  provided  that  Ave  regard  this  force  as  con- 
stant throughout  the  dimensions  of  the  body,  which  we  may 
do  in  all  practical  cases.  Hence,  the  centre  of  gravity  is  the 
same  for  the  same  body,  wherever  it  may  be  situated.  The 
determination  of  the  centre  of  gravity  is,  then,  reduced  to 
the  determination   of  the   centre  of  a   system   of  parallel 


CENTKE    OF    GRAVITT.  63 

forces.  Equations  (13)  are,  therefore,  immediately  appli- 
plicable. 

Preliminary  discussion. 

52.  Let  there  be  any  number  of  weights  applied  at 
points  of  a  straight  line.  We  may  take  the  axis  of  X.  to 
coincide  with  this  line,  and  because  the  points  of  application 
of  the  weights  are  on  this  line,  we  shall  have, 

V  —  °5   y'  —  0?  &c-  >     z  —  0,    z'  —  0,  &g.  ; 

substituting  these  in  the  second  and  third  of  Equations  (13), 
we  have, 

yx  —  0,   and   Sj  =  0. 

Hence,  the  point  of  application  of  the  resultant  is  on  the 
given  line. 

In  the  case  of  a  material  straight  line,  that  is,  of  a  line 
made  up  of  material  points,  the  weight  of  each  point  will  be 
applied  at  that  point,  and  from  what  has  just  been  shown, 
the  point  of  application  of  the  resultant  weight  will  also  be 
on  the  line  ;  but  this  point  is  the  centre  of  gravity  of  the  line. 

Hence,  the  centre  of  gravity  of  a  material  straight  line 
is  situated  somewhere  on  the  line. 

Let  weights  be  applied  at  points  of  a  given  plane.  We 
may  take  the  plane  XY  to  coincide  with  this  plane,  and  in 
this  case  we  shall  have, 

z  =   0,    z'  ~   0,    &c. ; 
these  in  the  third  of  Equations  (13)  will  give, 
Si  =  0; 

hence,  the  point  of  application  of  the  resultant  iceights  is 
in  the  plane. 

It  may  be  shown,  as  before,  that  the  centre  of  gravity  of 
<!  material  plane  curve,  or  of  a  material  plane  area,  is  in 
the  plane  of  the  curve,  or  area. 

If  the  bodies  considered  are  homogeneous  in  structure, 
the  weights  of  any  elementary  portions  are  proportional  to 


64 


MECHANICS. 


1 


B 

Fig.  31. 


*£. 


B'A'. 


their  volumes,  and  the  problem  for  finding  the  centre  of 
gravity  is  reduced  to  that  for  finding  the  centre  of  figure. 
In  what  follows,  lines  and  surfaces  will  be  considered  as  made 
up  of  material  points,  and  all  the  volumes  considered  will  be 
regarded  as  homogeneous  unless  the  contrary  is  stated. 

Centre  of  Gravity  of  a  straight  line. 

53.  Let  there  be  two  material 
points  M  and  N,  equal  in  weight, 
and  firmly  connected  by  an  inflexible 
line  MN~.  The  resultant  of  these 
weights  will  bisect  the  line  MN"  in  S 
(Art.  40) ;  hence  S  is  the  centre  of 
gravity  of  the  two  points  M  and  N. 

Let  il/iV"  be  a  material  straight  line,  and  S  its  middle 
point.  We  may  regard  it  as  com- 
posed of  heavy  material  points  A,  A' ; 
B,  B ',  &c,  equal  in  weight,  and  so 
disposed  that  for  each  point  on  one 
side  of  S,  there  is  another  point  on 
the  other  side  of  it  and  equally  distant 
from   it.    From   what    precedes,  the 

centre  of  gravity  of  each  pair  of  points  is  at  S,  and  conse- 
quently the  centre  of  gravity  of  the  whole  line  is  at  S. 
That  is,  the  centre  of  gravity  of  a  straight  line  is  at  its 
middle  point. 

Centre  of  Gravity  of  symmetrical  lines  and  areas. 

54.  Let  APBQ  be  a  plane  curve,  and  AB  a  diameter, 
that  is,  a  line  which  bisects  a  system  of 

parallel  chords;  let  PQ  be  one  of  the 
chords  bisected.  The  centre  of  grav- 
ity of  the  chord  PQ  will  be  upon  AB, 
and  in  like  manner,  the  centre  of 
gravity  of  any  pair  of  points  lying  at 
the  extremity  of  one  of  the  parallel 
chords  will  be  found  upon  the  diam- 
eter; hence,  the  centre  of  gravity  of  the  entire  curve  is  upon 
the  diameter  (Art.  52).     The   entire  area  of  the  curve  is 


Fig.  32. 


CENTRE   OF    GRAVITY.  65 

made  up  of  the  system  of  parallel  chords  bisected,  and  since 
the  centre  of  gravity  of  each  chord  is  upon  the  diameter,  it 
follows  that  the  centre  of  gravity  of  the  area  is  upon  the 
diameter. 

Hence,  if  any  curve,  or  area,  has  a  diameter,  the  centre 
of  gravity  of  the  curve,  or  area,  lies  upon  that  diameter. 

If  a  curve  or  area  has  two  diameters,  the  centre  of  gravity 
will  be  found  at  their  point  of  intersection.  Hence,  in  the 
circle  and  ellipse  the  centre  of  gravity  is  at  the  centre  of  the 
curve. 

If  a  surface  has  a  diametral  plane,  that  is,  a  plane  which 
bisects  a  system  of  parallel  chords  terminating  in  the  surface, 
then  will  the  centre  of  gravity  of  the  extremities  of  each 
chord  lie  in  the  diametral  plane,  and  consequently,  the  cen- 
tre of  gravity  of  the  surface  will  be  in  that  plane.  The 
centre  of  gravity  of  the  volume  bounded  by  such  a  surface, 
for  like  reason,  lies  in  the  diametral  plane. 

Hence,  if  a  surface,  or  volume,  has  a  diametral  p>lane,  the 
centre  of  gravity  of  the  surface,  or  volume,  lies  in  that 
plane.  If  a  surface,  or  volume,  has  three  diametral  planes 
intersecting  each  other  in  a  point,  that  point  is  the  centre 
of  gravity.  Hence,  the  centre  of  gravity  of  the  sphere  and 
the  ellipsoid  lie  at  their  centres.  We  see,  also,  that  the 
centre  of  gravity  of  a  surface,  or  volume,  of  revolution  lies 
in  the  axis  of  revolution. 

Centre  of  Gravity  of  a  Triangle. 
55.     Let  ABC  be  any  plane  triang'e.     Join  the  vertex 
A  with  the  middle  point  D  of  the  op- 
posite side  B  C ;  then  will  AD  bisect  -A- 
all  of  the  lines  drawn  in  the  triangle                       /l\ 
parallel  to  the  base  BC;    hence,  the                 T./  ;    \ 
centre  of  gravity  of  the  triangle  lies              /    \  7?.    \ 
upon  AD  (Art.  54) ;  for  a  like  reason,         »  r:":ss??£::::'-i«A 
the  centre  of  gravity  of  the  triangle                       D 

o  Fi"  34. 

lies  upon  the  line  BE,  drawn   from 

the  vertex  B  to  the  middle  point  of  the  opposite  side  AC ; 

it  is,  therefore,  at  G,  their  point  of  intersection. 


66  MECHANICS. 

Draw  ED ;  then,  since  ED  bisects  A  C  and  B  (7,  it  ib 
parallel   to   AB,   and    the    triangles 
EGD  and  AGB  are   similar.     The  f 

side  ED  is  equal  to   one-half  of  its  / \  \ 

homologous   side   ^4_Z?,   consequently  %^  /     \ 

the  side   GD  is  equal  to  one-half  of         /..h\t~--^\ 
its  homologous  side  AG  ;  that  is,  the     C^"         ^         ^b 
point  G  is  one-third  of  the  distance  Fig.  34. 

from  D  to  ^4. 

Hence,  the  centre  of  gravity  of  a  plane  triangle  is  on  a 
line  drawn  from  the  vertex  to  the  middle  point  of  the  base, 
and  at  one-third  of  the  distance  from  the  base  to  the  vertex. 

Centre  of  Gravity  of  a  Parallelogram. 

56.  Let  A  C  be  any  parallelogram.  Draw  EF  bisect- 
ing  the  sides  A B  and   CD ;   it  will 

also  bisect  all  lines  of  the  parallelo-  J) E  C 

gram  parallel  to  these  sides ;  hence,  the  /      ^i 

centre  of  gravity  lies  on  it ;  draw  also  /        T 

the  line  Oil  bisecting  the  sides  AD        j- =£— 

and  BC;   for   a   similar  reason,   the  Fig. 35. 

centre  of  gravity  lies  on  it :  it  is, 
therefore,  at  (r,  their  point  of  intersection. 

Hence,  the  centre  of  gravity  of  a  parallelogram  lies  at 
the  point  of  intersection  of  two  straight  lines  joining  the 
middle  points  of  the  opposite  sides. 

It  is  to  be  remarked,  that  this  point  coincides  with  the 
point  of  intersection  of  the  diagonals  of  the  parallelogram. 
Centre  of  Gravity  of  a  Trapezoid. 

57.  Let  AC  be  a  trapezoid.  Join  the  middle  points, 
0  and  I\  of  the  parallel   sides,  by  a 

straight  line ;  this  line  will  bisect  all  -JJl 

lines  parallel  to  AB  and  DC\  hence, 
it  must  contain  the  centre  of  gravity. 
Draw  the  diagonal  BI),  dividing  the 
trapezoid  into  two  triangles.  Draw 
also   the    lines   DO   and    BP\    take 


CENTRE    OF    GRAVITY. 


67 


OQ  =  \OD,  and  PB  =  ±PB  ;  then  will  Q  and  B  be 
the  centres   of  gravity  of  these  triangles  (Art.  55).     Join 

Q  and  B  by  a  straight  line ;  the  centre  of  gravity  of  the 
trapezoid  must  be  on  this  line  (Art.  52).  Hence,  it  is  at 
G  where  the  line  QB  cuts  OP. 

Centre  of  Gravity  of  a  Polygon. 

58.  Let  ABODE  be  any  polygon,  and  «,  6,  c,  <7,  e,  the 
middle  points  of  its  sides.  The  weights 
of  the  sides  will  be  proportional  to 
their  lengths,  and  may  be  represented 
by  them.  Let  it  first  be  required  to 
find  the  centre  of  gravity  of  the  peri- 
meter ;  join  a  and  b,  and  find  a  point 
o,  such  that 

ao  :  ob  :  :  BO  :  BA; 


then  will  o  be  the  centre  of  gravity  of  the  sides  AB  and 
B  O.     Join  o  and  c,  and  find  a  point  o',  such  that 


oo 


o  e 


CD  :  AB  +  BC\ 


then  will  o'  be  the  centre  of  gravity  of  the  three  sides, 
AB,  B  (7,  and  OD.  Join  o'  with  d,  and  proceed  as  before, 
continuing  the  operation  till  the  last  point,  G,  is  found  ;  this 
will  be  the  centre  of  gravity  of  the  perimeter. 

To  find  the  centre  of  gravity  of  the  area,  divide  it  into 
the  least  number  of  triangles  possible,  and  find  the  centre 
of  gravity  of  each  triangle.     The  weights  of  these  triangles 
will   be  proportional  to   their   areas, 
and    may   be   represented   by   them. 
(Art.  52*.)     Let  ABODE  A  he  any 
polygon,  aud  0,  0\  0",  the  centres  of 
gravity  of  the  triangles  into  which  it 
can  be  divided.     Join  0  and  0\  and 
find  a  point  0"\  such  that 


O'O' 


00"' 


ABO  :  AOD; 


Fig.  8S. 


68 


MKCIIANICS. 


then  will  0'"  be  the  centre  of  grav- 
ity of  the  two  triangles  ABC  and 
A  CD.  A 

Join  0"  and  0'",  and  find  a  point 
G,  such  that 


Fig.  38. 


0"'G  :  0"G  :  :  ABE  :  ABC  +  ACD; 

then  will  G  be  the  centre  of  gravity  of  the  given  polygon. 

Every  curvilinear  area  may  be  regarded  as  polygonal,  the 
number  of  sides  being  very  great.  Hence,  the  centres  of 
gravity  of  their  perimeters  and  areas  may  be  found  by  the 
methods  given. 

Centre  of  Gravity  of  a  Pyramid. 

59«  Any  triangular  pyramid  may  be  regarded  as  made 
up  of  infinitely  thin  layers  parallel  to  either  of  its  faces.  If 
a  straight  line  be  drawn  from  either  vertex  to  the  centre  of 
gravity  of  the  opposite  face,  it  will  pass  through  the  centres 
of  gravity  of  all  the  layers  parallel  to  that  face.  We  may 
regard  the  weight  of  each  layer  as  being  applied  at  its  cen- 
tre of  gravity,  that  is,  at  a  point  of  this  line  ;  hence,  the 
centre  of  gravity  of  the  pyramid  is  on  this  line  (Art.  52). 

Let  AB  CD  be  a  pyramid,  and  K  the  middle  point  of 
DC.  Draw  KB  and  KA,  and  lay- 
off KO  =  ±KB,  and  KO'  =  \KA. 
Then  will  0  be  the  centre  of  gravity 
of  the  face  DB  C,  and  0'  that  of  the 
face  CAD.  Draw  A  0  and  BO'  in- 
tersecting in  G.  Because  the  centre 
of  gravity  of  the  pyramid  is  upon  both 
A  0  and  B  0',  it  is  at  their  intersection 
G.  Draw  00';  then  KO  and  KO' 
being  respectively  third  parts  of  KB 
and  KA,  0  0'  is  parallel  to  AB,  and 
the  triangles   OGO  and  AGB  are  similar,  consequently 


CENTRE    OF    GRAVITY.  69 

their  homologous  sides  are  proportional.  But  00'  is  one- 
third  of  AJB,  consequently  OG  is  one-third  of  GA,  or  one- 
fourth  of  AO. 

Hence,  the  centre  of  gravity  of  a  triangular  pyramid  is 
on  a  line  drawn  from  its  vertex  to  the  centre  of  gravity  of 
its  base,  and  at  one-fourth  of  the  distance  from  the  base  to 
the  vertex. 

Either  face  of  a  triangular  pyramid  may  be  taken  as  the 
base,  the  opposite  vertex  being  considered  as  the  vertex  of 
the  pyramid. 

To  find  the  centre  of  gravity  of  a  polygonal  pyramid ;  let 
A-BCDEF,  represent  any  pyramid,  A  being  the  ver- 
tex. Conceive  it  divided  into  tri- 
angular pyramids,  having  a  common 
vertex  at  A.  If  a  plane  be  passed 
parallel  to  the  base,  and  at  one-fourth 
of  the  distance  from  the  base  to  the 
vertex,  it  follows,  from  what  has  just 
been  shown,  that  the  centres  of  gravi- 
ty of  all  the  partial  pyramids  will  lie 
in  this  plane.  We  may  regard  each 
pyramid  as  having  its  weight  concen- 
trated at  its  centre  of  gravity ;  hence,  the  centre  of  gravity 
of  the  entire  pyramid  must  lie  in  this  plane  (Art.  52).  But 
it  may  be  shown,  as  in  the  case  of  the  triangular  pyramid, 
that  the  centre  of  gravity  lies  somewhere  in  the  line  drawn 
from  the  vertex  to  the  centre  of  gravity  of  the  base  ;  it  must, 
therefore,  lie  where  this  line  pierces  the  auxiliary  plane : 

Hence,  the  centre  of  gravity  of  any  pyramid  whatever 
lies  on  a  line  drawn  from  its  vertex  to  the  centre  of  gravity 
of  its  base,  and  at  one-fourth  of  the  distance  from  the  base 
to  the  vertex. 

A  cone  is  a  pyramid  having  an  infinite  number  of  faces  : 

Hence,  the  centre  of  gravity  of  a  cone  is  on  a  line  drawn 
from  the  vertex  to  the  centre  of  gravity  of  the  base,  and  at 
one-fourth  of  the  distance  from  the  base  to  the  vertex. 


70  MECHANICS. 

Centre  of  Gravity  of  Prisms  and  Cylinders. 

60.  Any  prism  whatever  may  be  regarded  as  made  up 
of  layers  parallel  to  the  bases.  If  a  straight  line  be  drawn 
between  the  centres  of  gravity  of  the  two  bases,  it  will  pass 
through  the  centres  of  gravity  of  all  these  layers.  The 
centre  of  gravity  of  the  prism  will,  therefore,  lie  somewhere 
in  this  line,  which  we  may  call  the  axis  of  the  prism.  We 
may  also  regard  the  prism  as  made  up  of  material  lines 
parallel  to  the  lateral  edges  of  the  prism.  If  a  plane  be 
passed  midway  between  the  two  bases  and  parallel  to  them, 
it  will  bisect  all  of  these  lines,  and  consequently  their 
centres  of  gravity,  as  well  as  that  of  the  entire  prism, 
will  lie  in  it.  It  must,  therefore,  be  at  the  point  in  which 
the  plane  cuts  the  axis  of  the  prism,  that  is,  at  its  middle 
point. 

Hence,  the  centre  of  gravity  of  a  prism  is  at  the  middle 
point  of  its  axis. 

When  the  bases  of  the  prism  become  polygons  having  an 
infinite  number  of  sides,  the  prism  will  become  a  cylinder, 
and  the  principle  just  demonstrated  will  still  hold  good: 

Hence,  the  centre  of  gravity  of  a  cylinder  icith  parallel 
bases  is  at  the  middle  point  of  its  axis. 

Centre  of  Gravity  of  Polyhedrons. 

61.  If  any  point  within  a  polyhedron  be  assumed,  and 
this  point  be  joined  with  each  vertex  of  the  polyhedron,  we 
shall  thus  form  as  many  pyramids  as  the  solid  has  faces  :  the 
centres  of  gravity  of  these  pyramids  may  be  found  by  the 
rules  for  such  cases.  If  the  centres  of  gravity  of  the  first 
and  second  pyramid  be  joined  by  a  Straight  line,  the  com- 
mon centre  of  gravity  of  the  two  may  be  found  by  a 
process  entirely  similar  to  that  used  in  finding  the  centre  of 
gravity  of  a  polygon,  observing  that  the  weights  of  the  par 
tial  pyramids  are  proportional  to  their  volumes,  and  that 
they  may  be  represented  by  their  volumes.  Having  com- 
pounded the  weights  of  the  first  ami  second,  and  found  its 
point  of  application,  we  may,  in  like  manner,  compound  this 


CENTRE    OF    GRAVITY.  71 

with  the  weight  of  the  third,  and  so  on,  till  the  centre  of 
gravity  of  the  entire  pyramid  is  determined. 

Any  solid  body  bounded  by  a  curved  surface  may  be 
regarded  as  a  polyhedron  whose  faces  are  extremely  small, 
audits  centre  of  gravity  may  be  determined  by  the  rule  just 
explained. 

Experimental  determination  of  the  Centre  of  Gravity. 

63.  We  know  that  the  weight  of  a  body  always  passes 
through  its  centre  of  gravity,  no  matter  what  may  be  the 
position  of  the  body.  If  we  attach  a  flexible  cord  to  a  body 
at  any  point  and  suspend  it  freely,  it  must  ultimately  come 
to  a  state  of  rest.  In  this  position,  the  body  is  acted  upon 
by  two  forces:  the  weight,  tending  to  draw  the  body  towards 
the  centre  of  the  earth,  and  the  tension  of  the  cord,  which 
resists  this  force.  In  order  that  the  body  may  be  in  equili- 
brium, these  forces  must  be  equal  and  directly  opposed. 
But  the  direction  of  the  weight  passes  through  the  centre 
of  gravity  of  the  body ;  hence,  the  tension  of  the  string, 
which  acts  in  the  direction  of  the  string,  must  also  pass 
through  the  same  point.  This  principle  gives  rise  to  the 
following  method  of  finding  the  centre  of  gravity  of  a 
body. 

Let  AB  C  represent  a  body  of  any  form  whatever.  Attach 
a  string  to  any  point,  C,  of  the  body, 
and  suspend  it  freely  ;  when  the  body 
comes  to  a  state  of  rest,  mark  the  di- 
rection of  the  string  ;  then  suspend  the 
body  by  a  second  point,  JB,  as  before, 
and  when  it  comes  to  rest,  mark  the 
direction  of  the  string ;  their  point  of 
intersection,   G,  will  be   the  centre  of  _,    „ 

'         '  Fig.  41. 

gravity  of  the  body. 

Instead  of  suspending  the  body  by  a  string,  it  may  be 
balanced  on  a  point.  In  this  case,  the  weight  acts  vertically 
downwards,  and  is  resisted  by  the  reaction  of  the  point ; 
hence,  the  centre  of  gravity  must  lie  vertically  over  the 
point. 


72  MECHANICS. 

If,  therefore,  the  body  be  balanced  at  any  two  points  of 
its  surface,  and  verticals  be  drawn  through  the  point,  in 
these  positions,  their  intersection  will  be  the  centre  of  gravi- 
ty of  the  body. 

It  follows,  from  what  has  just  been  explained,  that  when 
a  body  is  suspended  by  an  axis,  it  can  only  come  to  a  state 
of  rest  when  the  centre  of  gravity  lies  in  a  vertical  plane 
passed  through  the  axis. 

The  centre  of  gravity  may  lie  above  the  axis,  below  the 
axis,  or  on  the  axis. 

In  the  first  case,  if  the  body  be  slightly  deranged,  it  will 
continue  to  revolve  till  the  centre  of  gravity  falls  below  the 
a\i>  ;  in  the  second  case,  it  will  return  to  its  primitive  po- 
sition ;  in  the  third  case,  it  will  remain  in  the  position  in 
which  it  is  placed.  These  cases  will  be  again  referred  to, 
under  the  head  of  Stability. 

The  preceding  rules  enable  us  to  find  the  centres  of  gravi- 
ty of  all  lines,  surfaces,  and  solids  ;  but,  on  account  of  the 
difficulty  of  applying  them  in  certain  cases,  we  shall  annex 
an  outline  of  some  of  the  methods,  by  the  Differential  and 
Integral  Calculus.  Those  magnitudes  whose  centres  of  grav- 
ity are  most  readily  found  by  the  calculus,  are  mathematical 
curves;  areas  bounded  wholly,  or  in  part,  by  these  curves  ; 
curved  surfaces ;  and  volumes  bound  by  curved  surfaces. 

Determination  of  the  Centre  of  Gravity  by  means  of  the  Calculus. 

6-1.  To  place  Formulas  (13)  under  a  suitable  form  for  the 
application  of  the  calculus,  we  have  simply  to  substitute  for 
the  forces  />,  7>,  <fcc,  the  elementary  volumes,  or  the  differ- 
entials of  the  magnitudes,  and  to  replace  the  sign  of  summa- 
tion,  2,   by  that  of  integration,  /. 

Making  these  changes,  Formulas  (13)  become, 

In  which  dm  denotes  the  differential  of  the  magnitude  in 
question  ;  a*,  y,  and  z,  the  co-ordinates  of  its  centre  of  grav- 


CENTRE    OF    GRAVITY. 


73 


ity,  and  a?15  y17  and  zn  the  co-ordinates  of  the  centre  of  grav- 
ity of  the  magnitude. 

Application  to  plane  curves. 
65.  The  plane  JTY  may  be  taken  to  coincide  with  that 
of  the  curve,  in  which  case,  z  =  0  for  every  point  of  the 
curve  ;  and,  consequently,  zx  =  0  ;  dm  becomes  the  differ- 
ential of  an  arc  of  a  plane  curve,  or  dm  =  y^sc3  +  dy*. 
Substituting  in  (32),  Ave  have, 


fxy/da?+  dy* 
f  i/dx*  4-  dy2 


Vi  = 


fyVdx^dy^ 
f^/dtf  +  dy* 


(33.) 


Centre  of  Gravity  of  an  arc  of  a  circle. 
66.  Let  AB  C  be  the  arc,  0  the  origin  of  co-ordinates 
and  centre  of  the  circle,  OJC  the  axis  of 
abscissas,  perpendicular  to  the  chord  of 
the  arc,  and  O^Fthe  axis  of  ordinates. 
Since  the  arc  is  symmetrically  situated 
with  respect  to  the  axis  of  JT,  the  centre 
of  gravity  is  somewhere  on  this  line 
(Art.  54) ;  consequently,  yl  =  0.  To  find 
a^,  we  have  the  equation  of  the  circle, 


Differentiating, 


x2  +  ya   =  r\ 


2xdx  +  2ydy  =  0  ;         .*.     dx* 


V 


dy> 


Substituting  in  the  first  of  Formulas  (33),    and  reducing, 
we  find, 

frdy 

rdy 


x,   = 


Integrating  both  numerator  and  denominator  between  the 
limits  y  —  —  Jc,    and  y  =  +  \t,   we  have, 

J  rdy  =  re  ; 


-J« 


u 


MECHANICS. 


and, 


rely  .     .  c 

°        =  r  sin-1  — 


/ 


r  sin  ■ 


—  c 
~2r 


=  arc  ABC. 


Hence,  by  substitution, 


a5i   = 


re 


arc  ABC 


or,  arc  ABC  :  c  :  :  r 


That  is,  the  centre  of  gravity  of  an  arc  of  a  circle  is  on 
the  diameter  which  bisects  its  chord,  and  its  distance  from 
the  centre  is  a  fourth  proportional  to  the  arc,  chord,  and 
radius. 

Application  to  Plane  areas. 

67.  Let  the  plane  XY  be  taken  to  coincide  with  that 
of  the  area.  We  shall  have,  as  before,  zx  =  0.  In  this 
case,  we  have  dm  =  ydx  ;  and,  consequently,  Formulas 
(32),  reduce  to 

__  fxydx  _  fjfdx 

Xl  ~   fydx  ■   and2/l  "    fydx 


(34.) 


Centre  of  Gravity  of  a  parabolic  area. 

6§.  Let  A  OB  represent  the  area, 
haying  its  chord  at  right  angles  to  the 
axis.  Let  0  be  the  origin  of  co-ordinates, 
taken  at  the  vertex,  and  let  the  axis  of  Jl 
coincide  with  the  axis  of  the  curve;  the 
value  of  yx  will,  as  before,  be  equal  to  0. 
To  find  the  value  of  xx,  we  have  the  equa- 
tion of  the  parabola, 

/ —      i 

y%  =  2px     .'.     y  =  ~/2p  .  x~ . 


Fig.  48. 


By  substitution  in  the  first  of  Formulas  (34),  we  have, 


/V  2p  .x2dx        far  dx 
fy'2p  .x2dx        fx'dx 


CENTRE    OF    GRAVITY. 


7* 


Integrating  between   the  limits  x  =  0,  and  x  =  a,  we 
have, 


and, 


hence, 


J  x2dx  =  fa  , 


/«* 


cfo 


K 


«!  =  fa. 


That  is,  *Ae  cew£re  q/*  gravity  of  a  segment  of  a  parabola 
is  on  its  axis,  and  at  a  distance  from  the  vertex  equal  to 
three-fifths  of  the  altitude  of  the  segment. 


Application  to  solids  of  revolution. 

69.  If  we  take  the  planes  XY  and  XZ  passing  through 
the  axis  of  revolution,  the  centre  of  gravity  will  lie  in  both 
these  planes,  therefore  yl  and  g,  will  both  be  0.  In  this 
case,  the  first  of  Formulas  (32)  will  be  sufficient. 

Since  the  axis  of  X  coincides  with  the  axis  of  revolution, 
dm  becomes  equal  to  <xy*dx.  Substituting  in  the  first  of 
Formulas  (32),  we  have, 


/  xy*dx 
fy'dx 


(35.) 


Centre  of  Gravity  of  a  semi-ellipsoid. 

70.     Let  the  semi-ellipse  A  CBy  be  re-      -A- 

volved  about  the  axis  OC;  it  will  gener- 
ate a  semi-ellipsoid  whose  axis  coincides 
with  the  axis  of  X.  Both  y^  and  zx 
being  0,  it  only  remains  to  find  the  value 
of  xv 


Fig.  44. 


76  MECHANICS. 

The  equation  of  the  ellipse  referred  to  its  centre,  is, 

in  which  a  and  b  are  the  semi-axes. 

Substituting,  in  Equation  (35),  we  have, 

f  —  (a'2x  —  xs)dx  f(a2x  —  x*)dx 


!>' 


f  -{a*  -  x*)dx  f  (a1  -  x>)dx 

Integrating  between  the  limits,   x  =  0,   and  x  —  a,   we 
have 

a 

y  (°^  -  ***  =  (2-4)  =  4 ; 


and, 


a 


0 

Substituting,  we  have, 

a*         2a*        3  3 

a,  = '-  =  -a  =   —  x  2a. 

1  4  3  8  16 

That  is,  the  centre  of  gravity  of  a  semi-prolate  spheroid 
of  revolution  is  on  its  axis  of  revolution,  and  at  a  distance 
from  the  centre  equal  to  three-sixteenths  of  the  major  axis 
of  the  generating  ellipse. 

The  examples  above  given  are  enough  to  indicate  the 
method  of  applying  the  calculus  to  the  determination  of  the 
centre  of  gravity. 

Centre  of  Gravity  of  a  system  of  bodies. 
71.     When  we  have  several  bodies,  and  it  is  required  to 
find  their  common  centre  of  gravity,  it  will,  in  general,  be 
found  most  convenient  to  employ  the  principle  of  moments. 


CENTRE    OF    GRAVITY.  77 

To  do  this,  we  first  find  the  centre  of  gravity  of  each  body 
separately,  by  the  rules  already  given.  The  weight  of  each 
body  may  then  be  regarded  as  a  force  applied  at  the  centre 
of  gravity  of  the  body.  The  weights  being  parallel,  we 
have  a  system  of  parallel  forces,  whose  points  of  application 
are  known.  If  these  points  are  all  in  the  same  plane,  we 
may  find  the  lever  arms  of  the  resultant  of  all  the  weights, 
with  respect  to  two  lines,  at  right  angles  to  each  other  in 
that  plane ;  and  these  will  make  known  the  point  of  applica- 
tion of  the  resultant,  or,  what  is  the  same  thing,  the  centre 
of  gravity  of  the  system.  If  the  points  are  not  in  the  same 
plane,  the  lever  arms  of  the  resultant  of  all  the  weights  may 
be  found,  with  respect  to  three  axes,  at  right  angles  to  each 
other ;  these  will  make  known  the  point  of  application  of  the 
resultant  weight,  or  the  required  position  of  the  centre  of 
gravity. 

MISCELLANEOUS      EXAMPLES. 

1.  Required  the  point  of  application  of  the  resultant  of 
three  equal  weights,  applied  at  the  three  vertices  of  a  plane 
triangle. 

SOLUTION. 

Let  ABC  (Fig.  34)  represent  the  triangle.  The  resul- 
tant of  the  weights  applied  at  B  and  C  will  be  applied  at 
J9,  the  middle  point  of  BC.  The  weight  acting  at  D  being 
double  that  at  A,  the  total  resultant  will  be  applied  at  G, 
making  GA  —  2  GB;  hence,  the  required  point  is  at  the 
centre  of  gravity  of  the  triangle. 

2.  Required  the  point  of  application  of  the  resultant  of 
a  system  of  equal  parallel  forces,  applied  at  the  vertices  of 
any  regular  polygon  ? 

Ans.     At  the  centre  of  gravity  of  the  polygon. 

3.  Parallel  forces  of  3,  4,  5,  and  6  lbs.,  are  applied  at  the 
successive  vertices  of  a  square,  whose  side  is  12  inches.  At 
what  distance  from  the  first  vertex  is  the  point  of  applica- 
tion of  their  resultant  ? 


78 


MECHANICS. 


SOLUTION. 

Take  the  sides  of  the  square  through  the  first  vertex  aa 
axes  of  moments  ;  call  the  side  through  the  first  and  second 
vertex  the  axis  of  JT,  and  that  through  the  first  and  fourth 
the  axis  of  Y.     We  shall  have  from  Formulas  (13), 

4  x  12  +  5  x  12 
y,  =  -6 =   6; 

..  6x12  +  5  X  12  22 

and  x,    = — 

1  18  3 

Denoting  the  required  distance  by  e?,  we  have, 


d  —    V^i2  +  1/\    —   9.475  in.     Arts. 

4.  Seven  equal  forces  are  applied  at  seven  of  the  vertices 
of  a  cube.  "What  is  the  distance  of  the  point  of  application 
of  their  resultant  from  the  eighth  vertex  ? 

SOLUTION. 

Take  the  eighth  vertex  as  the  origin  of  co-ordinates,  and 
the  three  edges  passing  through  it  as  axes  of  moments.  We 
shall  have  from  Equations  (13),  denoting  one  edge  of  the 
cube  by  a, 

x1  =  ±a,    yx  —  ±a,    and  zx  =   ia. 

Denoting  the  required  distance  by  d,  we  have, 

d  —    yjx*  +  yx*  +  zf  =  ±a  <f&.     Ans. 

5.  Two  isosceles  triangles  are  constructed  on  opposite 
sides  of  the  base  b,  having  altitudes  respectively  equal  to 
h  and  h\  h  being  greater  than  ti.  Where  is  the  centre  of 
gravity  of  the  space  lying   within  the    two  triangles? 

SOLUTION. 

It  must  lie  on  the  altitude  of  the  greater  triangle.  Take 
the  common  base  as  an  axis  of  moments;  then  will  the 
moments  of  the  trinngles  be,  respectively,  ±bh  x  JA,    and 


CENTRE    OF    GRAVITY.  79 

\bh'  x  }Ji' ;  and  from  the  first  of  Formulas  (13),  we  shall 
have, 

That  is,  the  required  centre  of  gravity  is  on  the  altitude 
of  the  greater  triangle,  at  a  distance  from  the  common  base 
equal  to  one-third  of  the  difference  of  the  two  altitudes. 

6.  When  is  the  centre  of  gravity  of  the  space  included 
between  two  circles  tangent  to  each  other  internally  ? 

SOLUTION. 

Take  their  common  tangent  as  an  axis  of  moments.  The 
centre  of  gravity  will  lie  on  the  common  normal,  and  its 
distance  from  the  point  of  contact  is  given  by  the  equation, 

<bt3  —  cr'3        r2  +  rr'  +  r"1 


<rr2  —  tit'2  r  +  r' 

1.  Let  there  be  a  square,  and  suppose  it  divided  by  its 
diagonals  into  four  equal  parts,  one  of  which  is  removed. 
Required  the  distance  of  the  centre  of  gravity  of  the  re- 
maining figure  from  the  opposite  side  of  the  square. 

Ans.  T7g-  of  the  side  of  the  square. 

8.  To  construct  a  triangle,  having  given  its  base  and 
centre  of  gravity. 

SOLUTION. 

Draw  through  the  middle  of  the  base,  and  the  centre  of 
gravity,  a  straight  line ;  lay  off  beyond  the  centre  of  gra- 
vity a  distance  equal  to  twice  the  distance  from  the  middle 
of  the  base  to  the  centre  of  gravity.  The  point  thus  found 
is  the  vertex. 

9.  Given  the  base  and  altitude  of  a  triangle.  Required 
the  triangle,  when  its  centre  of  gravity  is  perpendicularly 
over  the  extremity  of  the  base. 

]  0.     Three  men  carry  a  cylindrical  bar,  one  taking  hold 


80  MECHANICS. 

of  one  end,  and  the  others  at  a  common  point.  Required 
the  position  of  this  point,  in  order  that  the  three  may  sus- 
tain equal  portions  of  the  weight. 

Pressure  of  one  body  upon  another. 

72.  Let  i  be  a  movable  body 
pressed  against  a  fixed  body  B, 
and  touching  it  at  a  single  point. 
In  order  that  A  may  be  in  equi- 
librium, the  resultant  of  all  the 
forces  acting  upon  it,  including  its 
weight,  must  pass  through  the  point 
of  contact,  P' ;  otherwise  there  would 
be  a  tendency  to  rotation  about  P\  ™    „ 

"  r  lg.  45. 

which   would   be    measured   by   the 

moment  of  the  resultant  with  respect  to  this  point.  Fur- 
thermore, the  direction  of  the  resultant  must  be  normal  to 
the  surface  of  B  at  the  point  P',  else  the  body  A  would 
have  a  tendency  to  slide  along  the  body  7i,  which  tendency 
would  be  measured  by  the  tangential  component.  The 
pressure  upon  B  develops  a  latent  force  of  reaction,  which 
must  be  equal  and  directly  opposed  to  it.  The  resultant  of 
all  the  forces  must  be  equal  to  zero  (Art.  47).  That  i<, 
when  a  body,  resting  upon  another  and  acted  upon  by  any 
number  of  forces  is  in  equilibrium,  the  resultant  of  all  the 
forces  called  into  play  is  equal  to  0. 

If  all  the  forces  called  into  play  are  taken  into  account, 
the  alejebraic  sums  of  their  moments  with  r>^j>r-t  to  any 
three  rectangular  axes  will  be  separately  equal  to  0. 

Equations  (29)  and  (30)  arc,  then,  perfectly  general  in 
every  case  <>f  equilibrium,  provided  all  of  the  forces  called 
into  play  are  taken  into  account. 

Stable,  Unstable,  and  Indifferent  Equilibrium. 

?3.  A  body  is  in  stable  equilibrium  when,  on  being 
slightly  disturbed  from  its  state  of  rest    it  has  a  tendency  to 


STABILTT. 


81 


0» 


>fc 


SB 


Fig.  46. 


return  to  that  state.     This  will,  in  general,  be  the  case  when 

the  centre  of  gravity  of  the  body  is  at  its  lowest  point.     Let 

A  be  a  spherical  body  suspended  from 

an    axis     0,  about  which  it    is  free    to 

turn.     When  the  centre   of    gravity   of 

A    lies   vertically  below  the    axis,  it  is 

in    equilibrium,    for   the  weight    of  the 

body  is  exactly  counterbalanced  by  the 

resistance  of  the   axis.      Moreover,   the 

equilibrium  is   stable;    for   if  the  body 

be  deflected  to  A\  its  weight  tends  to 

restore  it  to  its  position  of  rest,  A.     The  measure  of  this 

tendency  is  W  X  OP,  that  is,  the  moment  of  the  weight  with 

respect  to  the  axis  0.     Under  the  action  of  the  force  W,  the 

body  will  return  to  A,  and,  passing  to  the  other  side  by 

virtue  of  its  inertia,  will  finally  come   to  rest  and  return 

again  to  A\  and  so  on,  till  after  a  few  vibrations,  when  it 

will  come  to  rest  at  A. 

A  body  is  in  unstable  equilibrium  when,  being  slightly 
disturbed  from  its  state  of  rest,  it  tends  to  depart  still  far- 
ther from  it.  This  will,  in  general,  be  the  case  when  the 
centre  of  gravity  of  the  body  occupies  its  highest  position. 

Let  A  be  a  sphere,  connected  by  an 
inflexible  rod  with  the  axis  0.  When 
the  centre  of  gravity  of  A  lies  verti- 
cally above  0,  it  will  be  in  unstable 
equilibrium  ;  for,  if  the  sphere  be  de- 
flected to  the  position  A\  its  weight 
will  act  with  the  lever  arm  OP  to  in- 
crease this  deflection.  The  motion  will 
continue  till,  after  a  few  vibrations,  it  comes  to  rest 
below  the  axis.  In  this  last  position,  it  will  be  in  stable 
equilibrium. 

A  body  is  in  indifferent,  or  neutral,  equilibrium  when  it 
remains  at  rest  wherever  it  may  be  placed.     This  will,  in 
general,  be  the  case  when  the  centre  of  gravity  continues  in 
the  same  horizontal  plane  on  being  slightlv  disturbed. 
4* 


0> 


v  1& 


4P 


Fijr.  47. 


82 


MECHANICS. 


Let  A  be  a  sphere,  supported 
by  a  horizontal  axis  OP  passing 
through  its  centre  of  gravity. 
Then,  in  whatever  position  it  may 
be  placed,  it  will  have  no  tendency 
to  change  this  position ;  it  is,  therefore,  in  indifferent,  or 
neutral  equilibrium. 

In  the  figure,  A,  B,  and  C  represent  a  cone  in  positions 
of  stable,  unstable,  and  indifferent  equilibrium. 


Fie.  48. 


Fig.  49. 

If  a  wheel,  or  other  solid,  be  mounted  on  a  horizontal  axis, 
about  which  it  is  free  to  turn,  the  centre  of  gravity  not  lying 
on  the  axis,  it  will  be  in  stable  equilibrium,  when  the  centre  of 
gravity  is  directly  below  the  axis ;  and  in  unstable  equi- 
librium when  it  is  directly  above  the  axis.  When  the  axis 
passes  through  the  centre  of  gravity,  it  will,  in  every  po- 
sition, be  in  neutral  equilibrium. 

We  infer,  then,  from  the  preceding  discussion,  that  when 
a  body  at  rest  is  so  situated  that  it  cannot  be  disturbed  from 
its  position  without  raising  its  centre  of  gravity,  it  is  in  a 
state  of  stable  equilibrium  ;  when  a  slight  disturbance  de- 
presses the  centre  of  gravity,  it  is  in  a  state  of  unstable  equi- 
librium; when  the  centre  of  gravity  remains  constantly  in  the 
same  horizontal  plane,  it  is  in  a  state  of  neutral  equilibrium. 

This  principle  holds  true  in  combinations  of  wheels,  as  in 
machinery,  and  indicates  the  importance  of  balancing  the 
elements,  so  that  their  centres  of  gravity  may  remain  as 
nearly  as  possible  in  the  same  horizontal  planes. 


STABILITY.  S3 

Stability  of  Bodies  on  Horizontal  Planes. 

7-J.  A  body  resting  on  a  horizontal  plane  may  touch  it 
in  one,  or  in  more  than  one  point.  a 

In  the  latter  case,  the  salient  poly-  /!  \\ 

£on,  formed  by  joining  the  extreme  / — /J  \  \— -r 7 

points  of  contact,  as  abed,  is  called           /     l/_\/     / 
the  polygon  of  support.  L - — / 

Fig.  50. 

When  the  direction  of  the  weight  of  the  body,  that  is,  the 
vertical  through  its  centre  of  gravity,  pierces  the  plane  within 
the  polygon  of  support,  the  body  is  stable,  and  will  remain  in 
equilibrium,  unless  acted  upon  by  some  other  force  than  the 
weight  of  the  body.  In  this  case,  the  body  will  be  most 
easily  overturned  about  that  side  of  the  polygon  of  support 
which  is  nearest  to  the  line  of  direction  of  the  weight.  The 
moment  of  the  weight,  with  respect  to  this  side,  is  called 
the  moment  of  stability  of  the  body.  Denoting  the  weight 
of  the  body  by  W,  the  distance  from  the  line  of  direction 
of  the  weight  to  the  nearest  side  of  the  polygon  of  support, 
by  r,  and  the  moment  of  stability  by  $,  we  have, 

S  =   Wr. 

The  moment  of  stability  is  equal  to  the  least  moment  of 
any  extraneous  force  which  is  capable  of  overturning  the 
body  in  any  direction.  TIip.  weight  of  the  body  remaining 
the  same,  its  stability  will  increase  with  r.  If  the  polygon 
of  support  is  a  regular  polygon,,  the  stability  will  be  great- 
est, other  things  being  equal,  when  the  direction  of  the 
weight  passes  through  its  centre.  The  area  of  the  polygon 
of  support  remaining  constant,  the  stability  will  be  greater 
as  the  polygon  approaches  a  circle.  The  polygon  of  support 
being  regular,  but  variable  in  area,  the  stability  will  increase 
as  this  area  increases.  Hence,  low  bodies  resting  on  ex- 
tended bases,  are,  other  things  being  equal,  more  stable  than 
high  bodies  resting  on  narrow  bases. 

"When  the  direction  of  the  weight  passes  without  the 
polygon  of  support,  the  body  is  unstable,  and  unless  sup 


84  MECHANICS. 

ported  by  some  other  force  than  the  weight,  it  will  overturn 
about  that  side  which  is  nearest  to  the  direction  of  the 
weight.  In  this  case,  the  product  of  the  weight  into  the 
shortest  distance  from  its  direction  to  any  side  of  the  poly- 
gon, is  called  the  moment  of  instability.  Denoting  this 
moment  by  7",  we  have,  as  before, 

1=--   Wr. 

The  moment  of  instability  is  equal  to  the  least  moment 
of  any  force  which  can  be  applied  to  prevent  the  body  from 
overturning. 

If  the  direction  of  the  weight  intersect  any  side  of  the 
polygon  of  support,  the  body  will  be  in  a  state  of  equili- 
brium bordering  on  rotation  about  that  side. 

The  stability  of  a  body  will  be  greater,  the  more  nearly 
the  resultant  of  all  the  forces  acting  upon  it,  including  its 
weight,  is  to  being  normal  to  the  bearing  Burface.  A 
maximum  stability  will  be  obtained,  other  things  being 
equal,  when  the  resultant  is  exactly  perpendicular  to  the 
bearing  surface.  These  principles  find  application  in  most 
of  the  arts,  but  more  especially  in  Engineering  and  Architec- 
ture. In  structures  of  all  kinds  intended  to  be  stable,  the 
foundation  should  be  as  broad  as  is  consistent  with  the  gen- 
eral design  of  the  work,  that  the  polygon  <>1*  support  may 
be  as  great  as  possible.  The  pieces  tor  transmitting  pres- 
sures should  be  so  combined  that  the  pressures  transmitted 
to  the  ultimate  polygons  of  support  should  be  as  nearly 
normal  to  the  bearing  surfaces  as  possible,  and  their  lines 
<•['  direction  should  pass  as  near  the  centres  of  the  polygons 
of  support  as  may  be.  The  same  principles  hold  good  at 
all  the  points  of  junction  between  pieces  employed  for 
transmitting  pressures.  Hence,  joints  should  be  made  as 
nearly  normal  to  th  <  a  as  possible. 

In  the  construction  of  machinery  the  preceding  principles 
apply.  Tie-  centres  of  gravity  of  the  rotating  pieces 
should  be  on  their  axes,  otherwise  there  will  result  an  irre- 
gularity   of  motion,   which,  besides   making    the   machine 


STABILITY. 


85 


work  imperfectly,  will  ultimately  destroy  the  parts  of  the 
machine  itself. 

In  loading  cars,  wagons,  <fec,  we  should  endeavor  to 
throw  the  centre  of  gravity  of  the  load  as  near  the  track 
as  possible.  This  is,  in  practice,  partially  effected  by  placing 
the  heavier  articles  at  the  bottom  of  the  load. 

It  is  needless  to  enumerate  the  multitudinous  applications 
of  the  principles  of  stability ;  they  are  of  continual  occur- 
rence in  the  daily  transactions  of  life. 

PRACTICAL     PROBLEMS     IN     CONSTRUCTION. 


1.  A  horizontal  beam  AD, 
which  sustains  a  load,  is  sup- 
ported upon  a  pivot  at  A,  and 
by  a  cord  DE,  the  point  E 
being  vertically  over  A.  Re- 
quired the  tension  of  the  cord 
DE,  and  the  vertical  pressure 
on  the  pivot  A. 

SOLUTION. 


-AJf 


DC 


Fig.   51. 


Denote  the  weight  of  the  beam,  together  with  its  load, 
by  W,  and  suppose  its  point  of  application  to  be  at  C. 
Denote  C  A  by^?,  and  the  perpendicular  distance  AF,  from 
A  to  DE,  hyp'.  Denote  also  the  tension  of  the  cord  by  t. 
If  we  regard  A  as  the  centre  of  moments,  we  shall  have,  in 
the  case  of  an  equilibrium, 


Wp  =  tp'; 


t  =W. 


P 


Or,  denoting  the  angles  EDA  by  a,  and  the  distance  AD 
by  b,  we  shall  have, 

P 


IV, 


bsin 


To  find  the  vertical  pressure  on  the  pivot  A,  resolve  the 
force  t  into  two  components,  respectively  parallel  and  per 


86  MECHANICS. 

pendicularto  A3.    We  shall  have  for  the  latter  component, 
denoted  by  t\ 

t'  =  t  sina  =  W%  ' 

o 

The  vertical  pressure  upon  A,  plus  the  weight  W,  must 
be  equal  to  this  value  of  t'.  Denoting  this  pressure  by  P, 
we  shall  have, 

P+TT=TPf;or,P=ir(f-1)  =  TF(V); 

When  DC  =  0  ;  or,  when  D  and  C  coincide,  the  vertical 
pressure  becomes  0. 

2.  A  rope  AD,  supports  a  pole,  DO,  of  uniform  thick- 
ness, one  end  of  which  rests  upon  a 
horizontal  plane,  and  from  the  other 
end  is  suspended  a  weight  W.  Re- 
quired the  tension  of  the  rope,  and 
the  thrust,  or  pressure,  on  the  pole, 
the  weight  of  the  pole  being  neg- 
lected. Fig  52 

SOLUTION. 

Denote  the  tension  of  the  rope  by  t,  the  pressure  on  the 
pole  by  p,  the  angle  ADO  by  a,  and  the  angle  ODW 
by/3. 

There  are  three  forces  acting  at  Z>,  which  hold  each  other 
in  equilibrium;  the  weight  W,  acting  downwards,  the  ten- 
sion of  the  rope  acting  from  7>,  towards  A,  and  the  thrust 
of  the  pole  acting  from  0  towards  D.  Lay  off  D<K  to 
represent  the  weight,  and  complete  the  parallelogram  of 
forces  doaD\  then  will  Da  represent  the  tension  of  the 
rope,  and  Do  the  thrust  on  the  pole. 

From  Art.  35,  we  have, 

•  „r  sin  3 

t        H    :   :   sin   -j    :   sm  a  •  \     t  ■=.    W  --. —  • 

sm  a 


STABILITY.  87 

We  have,  also,  from  the  same  principle, 

p  :    W :  :  sm(a  +  p)  :  sina;         .*.    p  =   W — ^ -• 

1  v  '  r  sin  a. 

If  the  rope  is  horizontal,  we  shall  have  a  =  90°  —  /3, 
which  gives, 

£  =  7Ftan,3  ,    and  p  —  ^  • 

1  coso 

3.  A  beam  AJS,  is  suspended  by  two  ropes  attached  at 
its  extremities,  and  fastened  to  pins  A  and  II  Required 
the  tensions  upon  the  ropes. 

SOLUTION. 

Denote  the  weight  of  the  beam 
and  its  load  by  W,  and  suppose  that 
C  is  the  point  of  application  of  this 
force.      Denote    the    tension    of  the  Fi<r  ^ 

rope  BH,  by  t,  and  that  of  the  rope 

FA,  by  I'.  The  forces  acting  to  produce  an  equilibrium, 
are  W,  t,  and  t ' .  The  plane  of  these  forces  must  be  verti- 
cal, and  further,  the  directions  of  the  forces  must  intersect 
in  a  point.  Produce  AF,  and  BIT,  till  they  intersect  in  I\, 
and  draw  KG\  lay  oif  KC,  to  represent  the  weight  of  the 
beam  and  its  load,  and  complete  the  parallelogram  of  forces, 
Kb  Cf ;  then  will  Kb  represent  t,  and  Kf  will  represent  t'. 
Denote  the  angle  CKB  by  a,  and  the  angle  CKF  by  (3. 
We  shall  have,  as  in  the  last  problem, 

Wit    :  :  sin(a  +  /3)  :  sin/3  ;         .'.     t  =  W  ™fc+£l 

sin  p 

And, 
W  :  V  :  :  sin(a  +  /3)  :  sina  ;  .-.      f  =  W  ^  (" 


sin  a 


4.  A  gate  All,  is  supported  at  0  upon  a  pivot,  and  at 
A  by  a  hinge,  attached  to  a  post  AB.  Required  the 
pressure  on  the  pivot,   and  also  the  tension  of  the  hinge. 


88 


MECHANICS. 


AfX 


m. 


a 

i 

i ! 

of 

-ic 

i 

I I 


SOLUTION. 

Denote  the  weight  of  the  gate  and 
its  load,  by  W.  Produce  the  vertical 
through  the  point  of  application  (7,  of 
the  force  W,  till  it  intersects  the  hori- 
zontal through  A  in  D,  and  draw  the 
line  DO.  Then  will  DA  and  DO 
represent  the  directions  of  the  requir- 
ed components  of  W.  Lay  off  Dc, 
to  represent  the  value  of  JV,  and 
complete  the  parallelogram  of  forces,  Dcoa  ;  then  will  Do 
represent  the  pressure  on  the  pivot  0,  and  Da  the  pres- 
sure on  the  hinge,  A.  Denoting  the  angle  oDc  by  a,  the 
pressure  on  the  pivot  byjo,  and  on  the  hinge  hjp\  we  shall 
have, 

W  ,     .         p 


0       -E 
Fig.  54. 


P 


cosa 


and  p' 


sinx 


If  we  denote  the  distance  OE  by  b,  and  the  distance  DE 

by  A,  we  shall  have, 

h  _    .  5 

cosa  =  — ,    and  sin  a  = 


y/b3  +  A2 


V^2  +  A2 


Hence, 


7' 


ir-/6a 


h 


A2          .    ,       ^  -y/F+A3 
— ,    andjtr  =  v- 


5.  Having  given  the  two 
rafters  AG  and  i?(7  of  a 
roof,  abutting  in  notches  of 
a  tie-beam  AJJ,  it  is  required 
to  find  the  pressure,  or  thrust, 
upon  the  rafters,  and  the  di- 
rection and  intensity  of  the 
pressure  upon  the  joints  at  the  tie-bea  a. 


Fig.  55. 


SOLUTION. 

Denote  the  weight  of  the  ratters  and  their  load  by  2«?; 
we  may  regard  this  weight  as  made  up  of  three  parts — a 


STABILITY.  89 

weight  tc,  applied  at  C,  and  two  equal  weights  ^10,  applied 
at  A  and  B  respectively.  Let  us  denote  the  halt'  span  AL 
by  s,  the  rise  CL  by  /*,  and  the  length  of  the  rafter  A  C  or 
CB  by  I.  Denote,  also,  the  pitch  of  the  roof  CBL  by  a, 
the  thrust  on  the  rafter  by  £,  and  the  resultant  pressure  at 
each  of  the  joints  A  and  B  by  p. 

Lay  off  Co  to  represent  the  weight  10,  and  complete  the 
parallelogram  of  forces  Cboa  ;  then  will  Ca  and  Cb  repre- 
sent the  thrust  upon  the  rafters ;  and,  since  the  figure  Cboa 
is  a  rhombus,  we  shall  have, 

.   .  .  w  wl 

t  since  =  ±w  .'.     t  =  ; =  — r  • 

2  2  sina         2A 

Conceive  the  force  t  to  be  applied  at  A,  and  resolve  it  into 
two  components  respectively  parallel  to  CL  and  LA  ;  we 
shall  have  for  these  components, 

.   .  ,  ,  ics 

t  sina  =  ±w.    ana    t  cosa  =  — T  • 
2    '  2h 

The  latter  component  gives  the  strain  on  the  tie-beam, 
AB. 

To  find  the  pressure  on  the  joint,  we  have,  acting  down- 
wards, the  forces  \w  and  Jto,  or  the  single  force  ic,  and,  act- 
ing from  L  towards  A,  the  force  —j ;  hence, 


If  we  denote  the  angle  DAE  by  /3,  we  shall  have  from 
the  right-angled  triangle  DAE, 

♦      a       DE        s 

The  direction  of  the  joint  should  be  perpendicular  to  that 
of  the  force  p,  that  is,  it  should  make  with  the  horizon  an 

angle  whose  tangent  equals  —  • 


90 


MECHANICS. 


6.  In  the  last  problem  suppose  the  rafters  to  aout  against 
the  wall.  Required  the  least  thickness  that  must  be  given 
to  the  wall  to  prevent  it  from  being  overturned. 

SOLUTION. 

Denote  the  entire  weight  thrown  upon  the  wall  by  to,  the 
length  of  that  portion  of  the  wall  which  sustains  the  pressure 
p  by  l\  its  height  by  h\  its  thickness  by  jc,  and  the  weight 
of  each  cubic  foot  of  the  material  of  the  wall  by  w' ;  then 
will  the  weight  of  this  part  of  the  wall  be  equal  to  w'h'l'x. 

ws 
The  force  — =-  acts  with  an  arm  of  lever  h'  to  overturn  the 
2h 

wall  about  its  lower  and  outer  edge  ;  this  force  is  resisted  by 
the  weight  w  +  w'h'l'x,  acting  through  the  centre  of  gravity 
of  the  Avail  with  a  lever  arm  equal  to  \x.  If  there  is  an 
equilibrium,  the  moments  of  these  two  forces  must  be  equal, 


that  is,  ^j  x  h' 


'w  -f  w'h'l'x)  -,   or 


wsh' 


wx  +  w'h'l'x1 


ICS 


Reducing,  we  have,  x1  -\ rrj7,x  =     ,T,7 

&'  w  hi  wlh 


or,   x  = 


w  I  u 

2w'h'l'        V  w' 


ws  w 

Jd'  +  4w"h'H'2 


7.  A  sustaining  wall  has  a  cross  section  in  the  form  of  a 
trapezoid,  the  face  upon  which  the 
pressure  is  thrown  being  vertical,  and 
the  opposite  face  having  a  slope  of 
six  perpendicular  to  one  horizontal. 
Required  the  least  thickness  that  must 
be  given  to  the  wall  at  the  top,  that 
it  may  not  be  overturned  by  a  hori- 
zont;d  pressure,  whose  point  of  appli- 
cation is  at  a  distance  from  the  bottom  of  the  wall  equal  to 
one-third  of  its  height. 

SOLUTION. 

Pass  a  plane  through  the  edge  A  parallel  to   the  face 
J5C7,  and  consider  a  portion  of  the  wall  whose  length  is  one 


STABILITY. 


91 


foot.  Denote  the  pressure  upon  this  portion  by  P,  the 
height  of  the  wall  by  6A,  its  thickness  at  the  top  by  a*,  and 
the  weight  of  a  cubic  foot  of  the  material  by  ic.  Let  fall 
from  the  centres  of  gravity  0  and  0'  of  the  two  portions, 
the  perpendiculars  OG  and  0 E\  and  take  the  edge  D  as 
an  axis  of  moments.  The  weight  of  the  portion  AB  CF  is 
equal  to  Qiohx,  and  its  lever  arm,  DG,  is  equal  to  h  +  \x. 
The  weight  of  the  portion  ADF'y&  Sich2,  and  its  lever  arm, 
DE,  is  \h.  In  case  of  an  equilibrium,  the  sum  of  the  mo- 
ments of  their  weights  must  be  equal  to  the  moment  of  P, 
whose  lever  arm  is  2h.     Hence, 


or, 
Whence, 


6ichx(h  +  ix)  -f  Swh2  X  §A  =   P  X  2/t ; 
Qichx  +  Zwx*  +  2ioh*  =   2  P. 


jc2  +  2hx 


2{P-wh*)m 

Sic  ' 


\ 


72(P 


ic/f) 


3w 


+  h* 


E 


8.  Required  the  conditions  of  stability  of  a 
square  pillar  acted  upon  by  a  force  oblique 
to  the  axis  of  the  pillar,  and  applied  at  the 
centre  of  gravity  of  the  pillar's  upper 
base. 

SOLUTION. 

Denote  the  intensity  of  the  oblique  force 
by  P,  its  inclination  to  the  vertical  by  a, 
the  length  or  breadth  of  the  pillar  by  2a, 
its  height  by  #,  and  the  weight  of  the  pillar  by  W.  Through 
the  centre  of  gravity  of  the  pillar  draw  the  vertical  A  Cy 
and  lay  off  A  C  equal  to  7F~;  prolong  PA  and  lay  off  ^1P  equal 
to  P;  complete  the  parallelogram  of  forces  ABJ)C,  and 
prolong  the  diagonal  till  it  intersects  MG  or  IIG  produced. 
If  the  point  F  falls  between  II  and  G,  the  pillar  will  be 
stable ;  if  it  falls  at  II,  it  will  be  indifferent ;  if  it  falls  with- 
out II,  it  will  be  unstable.     To  find  an  expression  for  the 


m    a 

Fie.  57. 


92  MECHANICS. 

distance  FG,  draw  DE  perpendicular  to  AG.      From  the 
similar  triangles  ADE  and  AEG,  we  have, 

AGx  DE 


AE  :  AG  :  :  DE  :  FG;        .:    EG- 

But  AG  = 
hence  we  have 


AE 

But  AG  =  as,   DE  =  Psina,   and   vl^7  =   W+  Pcosa, 


_,  ^  Pec  sina 

FG  = 


W  +  Pcosa  ' 


And,  smce  ^T^  equals  «,  we  have  the  following  condi- 
tions for  stability,  indifference  and  instability,  respectively, 


a  > 


a  = 


«  < 


Px  sii 


IF  +  Pcosa  ' 

Px  sina 
IK  +  Pcosa  ' 

Px  sina 
TF  +  Pcosa' 


If  we  denote  the  distance  FG  by  y,  and  the  weight  of  a 
cubic  foot  of  the  material  of  the  pillar  by  W,  we  shall  have, 
since  W '=  4a*xw, 

Psina  x 
4a2ivx  +  Pcosa 

If,  now,  we  suppose  the  intensity  and  direction  of  the 
force  P  to  remain  the  same,  whilst  x  is  made  to  assume 
every  possible  value  from  0  up  to  any  assumed  limit,  the 
value  of  y  will  undergo  corresponding  changes.  The  suc- 
cessive points  thus  determined  make  up  a  line  which  is 
called  the  line  of  resistance,  and  whose  equation  is  that  just 
deduced. 

If  the  pillar  is  made  up  of  nncemented  blocks,  it  will  re- 
main in  equilibrium  so  long  as  each  joint  is  pierced  by  the 
line  of  resistance,  provided  that  the  tangent  to  the  line  of 
resistance  makes  with  the  normal  to  the  joint  an  angle  less 
than  the  limiting  angle  of  resistance  (Art.  103). 


STABILITY.  93 

The  highest  degree  of  stability  will  be  attained  when  the 
line  of  resistance  is  normal  to  every  joint,  and  when  it 
passes  through  the  centre  of  gravity  of  each. 

9.  To  determine  the  conditions  of  equilibrium  and  sta- 
bility of  an  arch  of  uncemented  stones. 

SOLUTION. 

Let  MNLK  represent  half  of  an  arch  sustained  in  equi- 
librium by  a  horizontal   force  P, 
and   by   the  weight   of  the    arch-  ^ — 3t 

stones.     Through    the    centre    of  /^Mutr 

gravity  of  the  first  arch-stone  draw  /c    ^r 

a  vertical  line,  and  on  it  lay  off  a         /   TV 
distance   to   represent   the  weight        /  D/  / 

of  that  stone.     Prolong  the  direc-       K~T^I 

tion  of  P,   and  lay  off  a  distance  ^e-53- 

equal   to   the   horizontal   pressure. 

Complete  the  parallelogram  of  forces,  aobB,  and  draw  the 
diagonal  oB.  This  will  be  the  resultant  of  the  forces  com- 
bined. Combine  this  resultant  with  the  weight  of  the 
second  arch-stone,  and  this  with  the  weight  of  the  third, 
and  so  on,  till  the  last  inclusive.  The  polygon  oBCDE, 
thus  found,  is  the  line  of  resistance,  and  if  this  lies  wholly 
within  the  solid  part  of  the  arch,  the  arch  will  be  stable  ; 
but,  if  it  does  not  lie  within  it,  the  arch  will  be  unstable. 
A  rupture  will  take  place  at  the  joint  where  the  line  of  re- 
sistance passes  without  the  solid  part  of  the  arch. 

This  problem  may  be  solved  analytically,  in  accordance 
with  the  principles  already  illustrated.  It  is  only  intended 
to  indicate  the  general  method  of  proceeding. 


94 


MECHANICS. 


CHAPTER  IV. 

ELEMENTARY       MACHINES. 

Definitions  and  General  Principles. 

75.  A  machine  is  a  contrivance  by  means  of  which  a 
force  applied  at  one  point  is  made  to  produce  an  effect  at 
6ome  other  point. 

The  force  applied  is  called  the  power,  and  the  point  at 
which  it  is  applied,  is  called  the  point  of  application,  The 
force  to  be  overcome  is  called  the  resistance,  and  the  point 
at  which  it  is  to  be  overcome  is  called  the  'working point. 

The  working  of  any  machine  requires  a  continued  applica- 
tion of  power.     The  source  of  this  power  is  called  the  motor. 

Motors  are  exceedingly  various.  Some  of  the  most  im- 
portant are  muscular  effort,  as  exhibited  by  man  and  beast 
in  various  kinds  of  work ;  the  weight  and  living  force  of 
water,  as  exhibited  in  the  various  kinds  of  water-mills ;  the 
expansive  force  of  vapors  and  gases,  as  displayed  in  steam 
and  caloric  engines;  the  force  of  air  in  motion,  as  exhi- 
bited in  the  windmill,  and  in  the  propulsion  of  sailing 
vessels ;  the  force  of  magnetic  attraction  and  repulsion,  as 
shown  in  the  magnetic  telegraph  and  various  magnetic 
machines;  the  elastic  force  of  springs,  as  shown  in  watches 
and  various  other  machines.  Of  these  motors,  the  most 
important  ones  are  steam,  air,  and  water  power. 

To  icorlc,  is  to  exert  a  certain  pressure  through  a  certain 
distance.  The  measure  of  the  quantity  of  work  performed 
by  any  force,  is  the  product  obtained  by  multiplying  the 
effective  pressure  exerted,  by  the  distance  through  which  it 
is  exerted. 

Machines  serve  simply  to  transmit  and  modify  the  action 
of  forces.     They  add  nothing  to  the  work  of  the  motor;  on 


ELEMENTARY    MA0H1NE8.  95 

the  contrary,  they  absorb  and  render  inefficient  much  of  the 
work  that  is  impressed  upon  them.  For  example,  in  the 
case  of  a  water-mill,  only  a  small  portion  of  the  work  ex- 
pended by  the  motor  is  transmitted  to  the  machine,  on 
account  of  the  imperfect  manner  of  applying  it,  and  of  this 
portion  a  very  large  fraction  is  absorbed  and  rendered  prac- 
tically useless  by  the  various  resistances,  so  that,  in  reality, 
only  a  small  fractional  portion  of  the  work  expended  by  the 
motor  becomes  effective. 

Of  the  applied  vjork,  a  part  is  expended  in  overcoming 
friction,  stiffness  of  cords,  bands,  or  chains,  resistance  of 
the  air,  adhesion  of  the  parts,  &q.  This  goes  to  wear  out 
the  machine.  A  second  portion  is  expended  in  overcoming 
sudden  impulses,  or  shocks,  arising  from  the  nature  of  the 
work  to  be  accomplished,  as  well  as  from  the  imperfect  con- 
nection of  the  parts,  and  from  the  want  of  hardness  and 
elasticity  in  the  connecting  pieces.  This  also  goes  to  strain 
and  wear  out  the  machine,  and  also  to  increase  the  sources 
of  waste  already  mentioned.  There  is  often  a  waste  of 
work  arising  from  a  greater  supply  of  motive  power  than  is 
required  to  attain  the  desired  result.  Thus,  in  the  move- 
ment of  a  train  of  cars  on  a  railroad,  the  excess  of  the  work 
of  the  steam,  above  what  is  just  necessary  to  bring  the  train 
to  the  station,  is  wasted,  and  has  to  be  consumed  by  the 
application  of  brakes,  an  operation  which  not  only  wears  out 
the  brakes,  but  also,  by  creating  shocks,  injures  and  ulti- 
mately destroys  the  cars  themselves. 

Such  are  some  of  the  sources  of  the  loss  of  work.  A 
part  of  these  may,  by  judicious  combinations  and  appliances, 
be  greatly  diminished ;  but,  under  the  most  favorable  cir- 
cumstances, there  must  be  a  continued  loss  of  work,  which 
requires  a  continued  supply  of  power  from  the  motor. 

In  any  machine,  the  quotient  obtained  by  dividing  the 
quantity  of  usefid,  or  effective  work,  by  the  quantity  of 
applied  work,  is  called  the  modulus  of  the  machine.  As  the 
resistances  are  diminished,  the  modulus  increases,  and  the 
machine  becomes  more  perfect.     Could  the  modulus  ever 


96  MECHANICS. 

become  equal  to  1,  the  machine  would  be  absolutely  perfect 
Once  set  in  motion,  it  would  continue  to  move  forever, 
realizing  the  solution  of  the  problem  of  perpetual  motion. 
It  is  needless  to  state  that,  until  the  laws  of  nature  are 
changed,  no  such  realization  need  be  looked  for. 

In  studying  the  principles  of  machines,  we  proceed  by 
approximation.  For  a  first  result,  it  is  usual  to  neglect  the 
effect  of  hurtful  resistances,  such  as  friction,  adhesion,  stiff- 
ness of  cords,  &c  Having  found  the  relations  between  the 
power  and  resistance  under  this  hypothesis,  these  relations 
are  afterwards  modified,  so  as  take  into  account  the  various 
resistances.  We  shall,  therefore,  in  the  first  instance,  regard 
cords  as  destitute  of  weight  and  thickness,  perfectly  flexible, 
and  inextensible.  We  shall  also  regard  bars  and  connecting 
pieces  as  destitute  of  weight  and  inertia,  and  perfectly  rigid ; 
that  is,  incapable  of  compression  or  extension  by  the  forces 
to  which  they  may  be  subjected. 

Elementary  Machines. 

76.  The  elementary  machines  are  seven  in  number — 
viz.,  the  cord  ;  the  lever  ;  the  inclined  pAane  ;  the  jndley,  a 
combination  of  the  cord  and  lever;  the  wheel  and  axle,  also  a 
combination  of  the  cord  and  lever;  the  screic,  a  combination 
of  two  inclined  planes  twisted  about  an  axis  ;  and  the  icedge, 
a  simple  combination  of  two  inclined  planes.  It  may  easily 
be  seen  that  there  are  in  reality  but  three  elementary 
machines — the  cord,  the  lever,  and  the  inclined  plane.  It 
is,  however,  more  convenient  to  consider  the  seven  above- 
named  as  elementary.  By  a  suitable  combination  of  these 
seven  elements,  the  most  complicated  pieces  of  mechanism 

are  produced. 

The  Cord. 

77,  Let  AB  represent  a  cord  solicited  by  two  forces, 
P  and  7?,  applied  at  its  extremi- 
ties, A   and  B.     In   order  that       p~*         ^  5        """*""* 

the  cord  may  be  in  equilibrium,  Flg  69 

it  is  evident,  in  the  first  place, 

that  two  forces  must  act  in  the  direction  of  the  cord,  and  in 


ELEMKNTARY    MACHINES.  97 

such  a  manner  as  to  stretch  it,  otherwise  the  cord  would 
bend  under  the  action  of  the  forces.  In  the  second  place, 
the  intensities  of  the  forces  must  be  equal,  otherwise  the 
greater  force  would  prevail,  and  motion  would  ensue. 
Hence,  in  order  that  two  forces  applied  at  the  extremities 
of  a  cord  may  be  in  equilibrium,  the  forces  must  be  equal 
and  directly  opposed. 

The  measure  of  the  tension  of  the  cord,  or  the  force  by 
which  any  tico  of  its  adjacent  particles  are  urged  to  sepa- 
rate, is  the  intensity  of  one  of  tJie  equal  forces,  for  it  is 
evident  that  the  middle  point  of  the  cord  might  be  fixed  and 
either  force  withdrawn,  without  diminishing  or  increasing 
the  tension.  When  a  cord  is  solicited  in  opposite  directions 
by  unequal  forces  directed  along  the  cord,  the  tension  will 
be  measured  by  the  intensity  of  the  lesser  force. 

Let  AB  represent  a  cord  solicited  by  two  groups  of  forces 
applied  at  its  two  extrem- 
ities. In  order  that  these 
forces  may  be  in  equilibrium, 
the  resultant  of  the  group  ap- 
plied at  A  and  the  resultant  of 

the  group  at  B  must  be  equal  and  directly  opposed.  Hence, 
if  we  suppose  all  of  the  forces  at  each  point  to  be  resolved  into 
components  respectively  coinciding  with,  and  at  right  angles 
to  AB,  the  normal  components  at  each  of  the  points  must 
be  such  as  to  maintain  each  other  in  equilibrium,  and  the 
resultants  of  the  remaining  components  at  each  of  the  points 
A  and  B  must  be  equal  and  directly  opposed. 

Let  ABCD  represent  a  cord,  at  the  different  points 
A,  B,  C,  D,  of  which  are 
applied  groups  of  forces.  If 
these  forces  are  in  equili- 
brium through  the  interven- 
tion of  the  cord,  there  must 
necessarily    be     an     equili-  * 

brium  at  each  point  of  ap- 
plication.    Denote  the  tension  of  AB,  BC,  CJJ,  by  t,  t',  t", 
5 


98  MECHANICS. 

and  the  forces  .applied  by  P,  P',  P",  <fcc,  as  shown  in  the 
figure.  The  forces  in  equilibrium  about  the  point  A  are 
P,  P',  P",  and  t,  directed  from  A  to  B  ;  the  forces  in  equili- 
brium about  B  are  P"\  PIT,  t,  directed  from  B  to  A,  and 
t\  directed  from  B  to  C.  The  tension  t  is  the  same  at  all 
points  of  the  branch  AB,  and,  since  it  acts  at  A  in  the  direc- 
tion AB,  and  at  B  in  the  direction  BA,  it  follows  that 
these  two  forces  exactly  counterbalance  each  other.  If, 
therefore,  the  forces  P,  P',  P',  were  transferred  from  A  to 
B,  unchanged  in  direction  and  intensity,  the  equilibrium  at 
that  point  would  be  undisturbed.  In  like  maimer,  it  may 
be  shown  that,  if  all  the  forces  now  applied  at  B  be  trans- 
ferred to  (7,  without  change  of  direction  or  intensity,  the 
equilibrium  at  C  would  be  undisturbed,  and  so  on  to  the 
last  point  of  the  cord.  Hence  we  conclude,  that  a  system  of 
forces  applied  in  any  manner  at  different  points  of  a  cord 
will  be  in  equilibrium,  when,  if  applied  at  a  single  point 
without  change  of  intensity  or  direction,  they  will  maintain 
each  other  in  equilibrium. 

Hence,  we  see  that  cords  in  machinery  simply  serve  to 
transmit  the  action  of  forces,  without  in  any  other  manner 
modifying  their  effects. 

The  Lever. 

7§.  A  lever  is  an  inflexible  bar,  free  to  turn  about  an 
axis.     This  axis  is  called  the  fulcrum. 

Levers  are  divided  into  three  classes,  according  to  the 
relative  positions  of  the  points  of  application  of  the  power 
and  resistance. 

In  the  first  class,  the  resist  a  nee  is 
beyond  both  the  power  and  fulcrum, 
an<l  on  the  s:de  of  the  fulcrum.     The 


common  weighing-scale  is  an  example 

of  this  class  of  levers.     The  matter  to 

be     weighed    is    the    resistance,    the 

counterpoising  weight  is   the   power,  Fig  62 

and   the   axis    of   suspension    is    the 

fulcrum. 


ELEMENTARY    MACHINES. 


99 


t 


2nd  Class. 


Fig.  63. 


8kd  Class. 


J 


In  the  second  class,  the  resistance 
is  between  the  power  and  the  ful- 
crum. The  oar  used  in  rowing  a 
boat  is  an  example  of  this  class  of 
levers.  The  end  of  the  oar  in  the 
water  is  the  fulcrum,  the  point  at 
which  the  oar  is  fastened  to  the  boat 
is  the  point  of  application  of  the  resist- 
ance, and  the  remaining  end  of  the  oar 
is  the  point  of  application  of  the 
power. 

In  the  third  class,  the  resistance  is 
beyond  both  the  fulcrum  and  the 
power,  and  on  the  side  of  the  power. 
The  treadle  of  a  lathe  is  an  example 
of  a  lever  of  this  kind.  The  point  at 
which  it  is  fastened  to  the  floor  is  the 
fulcrum,  the  point  at  which  the  foot  is 
applied  is  the  point  of  application  of 
the  power,  and  the  point  where  it  is 
attached  to  the  crank  is  the  point  of  application  of  the 
resistance. 

Levers  may  be  either  curved  or  straight,  and  the  direc- 
tions of  the  power  and  resistance  may  be  either  parallel  or 
oblique  to  each  other.  We  shall  suppose  the  power  and 
resistance  to  be  situated  in  planes  at  right  angles  to  the  ful- 
crum ;  for,  if  they  were  not  so  situated,  we  might  conceive 
each  to  be  resolved  into  two  components — one  at  right 
angles,  and  the  other  parallel  to  the  axis.  The  latter  com- 
ponent would  be  exerted  to  bend  the  lever  laterally,  or  to 
make  it  slide  along  the  axis,  developing  only  hurtful  resist- 
ance, whilst  the  former  only  would  tend  to  turn  the  lever 
about  the  fulcrum. 

The  perpendicular  distances  from  the  fulcrum  to  the  lines 
of  direction  of  the  power  and  resistance,  are  called  the  lever 
arms  of  these  forces.     In  the  bent  lever  MFN,  the  perpen* 


Fig.  64. 


100  MECHANIC8. 

dicular  distances  FA  and  FJB  are,  respectively,  the  lever 
arms  of  P  and  P. 

To  determine  the  conditions  of 
equilibrium  of  the  lever,  let  us 
denote  the  power  by  P,  the  re- 
sistance by  P,  and  their  respec- 
tive lever  arms  by  p  and  r.  We 
have  the  case  of  a  body  restrained  —    65 

by  an  axis,  and  if  we  take  this  as 
the  axis  of  moments,  we  shall  have  for  the  condition  of 
equilibrium  (Art.  49), 

Pp  =  Pr;    or,  P  :  P  :  ;  r  :  p  .     .     ( 36.) 

That  is,  the  power  is  to  the  resistance,  as  the  lever  arm  of 
the  resistance  is  to  the  lever  arm  of  the  power. 

This  relation  holds  good  for  every  kind  of  lever. 

The  ratio  of  the  power  to  the  resistance  when  in  equili 
brium,  either  statical  or  dynamical,  is  called  the  leverage,  oi 
mechanical  ad  van  tage. 

When  the  power  is  less  than  the  resistance,  there  is  said 
to  be  a  gain  of  power,  but  a  loss  of  velocity  ;  that  is,  the 
space  pulsed  over  by  the  power  in  performing  any  work,  is 
as  many  times  greater  than  that  passed  over  by  the  resis- 
tance, as  the  resistance  is  greater  than  the  power.  When 
the  power  is  greater  than  the  resistance,  there  is  said  to  be 
a  loss  of  power,  hut  <i  gain  of  velocity.  When  the  power 
and  resistance  are  equal,  there  is  neither  gain  nor  loss  of 
power,  but  simply  a  change  of  direction. 

In  levers  of  the  first  class,  there  may  be  either  a  gain  or 
a  loss  of  power  ;  in  those  of  the  second  class,  there  is  always 
a  gain  of  power;  in  those  of  the  third  class,  there  is  always 
a  loss  of  power.  A  gain  of  power  is  always  attended  with 
a  corresponding  loss  of  velocity,  and  the  reverse. 

If  several  forces  act  upon  a  lever  at  dim-rent  points,  all 
being  perpendicular  to  the  direction  of  the  fulcrum,  they 
will  be  in  equilibrium,  when  the  algebraic  sum  of  their 
moments,  with  respject  to  the  fulcrum,  is  equal  to  0. 


T 

A           B.i 

T 

Lc 

I 

l 

\ 

•p 

ELEMENTARY    MACHINES.  101 

This  principle  enables  ns  to  take  into  account  the  weight 
of  the  lever,  which  may  be  regarded  as  a  vertical  force 
applied  at  the  centre  of  gravity. 

The  pressure  on  the  fulcrum  is  equal  to  the  resultant  of 
the  power  and  resistance,  together  with  the  weight  of  the 
lever,  when  that  is  considered,  and  it  may  be  found  by  the 
rule  for  finding  the  resultant  of  forces  applied  at  points  of  a 
rigid  body. 

The  Compound  Lever. 

"79.  A  compound  lever  consists  of  a  combination  of 
simple  levers  AB,  B  C,  CD, 
so  arranged  that  the  resis- 
tance in  one  acts  as  a  power 
in  the  next,  throughout  the 
combination.  Thus,  a  power 
B  produces  at  B  a  resis- 
tance B',  which,  in  turn, 
produces  at  C  a  resistance  Fig.  66. 

B",  and  so  on.     Let  us  as- 
sume the  notation  of  the  figure.     From  the  principle  of  the 
simple  lever,  we  shall  have  the  relations, 

Pp  =  B'r",    By  =  B"r',    B"p"  =  Br. 

Multiplying  these  equations  together,  member  by  member, 
and  striking  out  the  common  factors,  we  have, 

Ppp'p"  —  Brr'r" ;    or,  P  :  B  :  :  rr'r"  :  pp'p".     ( 37.) 

AVe  might  proceed  in  a  similar  manner,  were  there  any 
number  of  levers  in  the  combination. 

Hence,  in  the  compound  lever,  the  power  is  to  the  resis- 
tance as  the  continued  product  of  the  alternate  arms  of 
lever,  commencing  at  the  resistance,  is  to  the  continued  pro- 
duct of  the  alternate  arms  of  lever,  commencing  at  the 
power. 

By  suitably  adjusting  the  simple  levers,  any  amount  of 
mechanical  advantage  may  be  obtained. 


102  MECHANICS. 

The  following  combination  is  used  where  a  great  pressure 
is  to  be  exerted  through  a  very  small  distance  : 

The  Elbow-joint  Press. 
80.     Let  CA,  BB,  and  BE  represent  bars,  with  hinge 
joints  at  B  and  B.     The 

bar   CA,  has  a  fulcrum  at  A- 

C,  and  the  bar  BE  works  /^^^<>^  v,      ^'\ 

through  a  guide  between  j  ^^^^5s>^  'v 

B   and  E.  ^  When   A    is        ?  |f}i^~[fe^— ^^^G 

do]  tressed,   BE   is   forced  | 

against  the  upright  F,  so  Fig.  67. 

as  to  compress,  with  great 

force,  any  body  placed  between  E  and  F.     This  machine  is 

called   the   elbow-joint  jyress,   and  is  used   in   printing,   in 

moulding  bullets,  in  striking  coins  and  medals,  in  punching 

holes,  riveting  steam  boilers,  <fcc. 

Let  P  denote  the  force  applied  at  A,  perpendicular  to 
AC,  Q  the  resistance  in  the  direction  BB,  and  R  the  com- 
ponent of  Q,  in  the  direction  EB.  Let  C  be  taken  as  an 
axis  of  moments,  and  then,  because  P  and  Q  are  in  equili- 
brium, we  shall  have, 

P  x  AC  =  Q  x  FC,   or,C=Px~. 

If  Ave  draw  BH  perpendicular  to  BR,  we  shall  have, 
cos  BBII  =  -jyn  i  but  we  have,  for  the  component  R, 

R  =  QcosBBTT  =  Q  x  ~^- 

Substituting  for  Q  its  value,  and  reducing, 

R  _    AC       BIT 
P ~   EC  *    BB ' 

When  B  is  depressed,  BTI  and  BB  approach  equality, 
and  EC  continually  diminishes;  that  is,  the  mechanical  ad- 
vantage increases,  and  finally,  when  B  reaches  ER,  it 
becomes  infinite.  There  is  no  limit  to  the  pressure  exerted 
at  E,  exeept  that  fixed  by  the  strength  of  the  machine. 


ELEMENTARY    MACHINES.  103 

The  Balance. 

81.     A  Balance  is  a  machine  for  weighing  hodies :  it 
consists  of  a  lever  AB,  called  the 
beam,  a  knife-edge  fulcrum  JFJ  and 

two  scale-pans  D  and  E,  suspended 
by  knife-edges  from  the  extremities 
of  the  lever  arms  FB  and  FA. 
These  arms  should  be  symmetrical, 
and  of  equal  length ;  the  knife- 
edges  A,  .Z?,  and  F,  should  all  lie 
in  the  same  plane,  and  be  perpen-  Fig.  68. 

dicular   to    a  plane   through   their 

middle  points  and  the  centre  of  gravity  of  the  beam ;  they 
are,  therefore,  parallel  to  each  other.  This  condition  of 
j:>arailelism  in  the  same  plane,  is  of  essential  importance. 

In  addition  to  this,  the  middle  points  of  the  knife-edges  A, 
B,  and  F,  should  be  on  the  same  straight  line,  perpendicular 
to  the  plane  through  the  fulcrum  F,  and  the  centre  of  gravity 
of  the  beam.  The  knife-edges  should  be  of  hardened  steel, 
and  their  supports  should  either  be  of  polished  agate,  or, 
what  is  still  better,  of  hardened  steel,  so  as  to  diminish  the 
effect  of  friction  along  the  lines  of  contact.  The  fulcrum 
may  be  made  horizontal,  by  leveling-screws  passing  through 
the  foot-plate  L.  A  needle  JV,  projects  upwards,  or  some- 
times downwards,  which,  playing  in  front  of  a  graduated 
arc  Gil,  serves  to  show  the  deflection  of  the  line  of  knife- 
edges  from  the  horizontal.  When  the  instrument  is  not  in 
use,  the  fulcrum  may  be  raised  from  its  bearings  by  a  pinion 
K,  working  into  a  rack  in  the  interior  of  the  standard  FK. 
The  knife-edges  A  and  B  may,  by  a  similar  arrangement, 
be  raised  from  their  bearings  also. 

The  ordinary  balances  of  the  shops  are  similar  in  their 
general  plan  ;  but  many  of  the  preceding  arrangements  are 
omitted.  The  scale-pans  being  exactly  alike,  the  balance 
should  remain  in  equilibrium,  with  the  line  AB  horizontal, 
not  only  when  the  balance  is  without  a  load,  but  also  when 
the  pans  are  loaded  with  equal  weights ;  and  when  AB  in 


10-i  MECHANICS. 

deflected  from  the  horizontal,  it  should  return  to  this  posi- 
tion. This  result  is  attained  by  throwing  the  centre  of 
gravity  slightly  below  the  line  A  B.  To  test  a  balance,  let 
two  weights  be  placed  in  the  pans  that  will  exactly  counter- 
balance each  other,  then  change  the  weights  to  the  opposite 
pans  ;  if  the  equilibrium  is  still  maintained,  the  balance  is 
said  to  be  true. 

The  sensibility  of  a  balance  is  its  capability  of  mdicatbig 
small  differences  of  weight.  The  sensibility  will  be  greater, 
as  the  lengths  of  the  arms  increase,  as  the  centre  of  gravity 
of  the  beam  approaches  the  fulcrum,  as  the  mass  of  the 
load  decreases,  and  as  the  length  of  the  needle  increases. 
The  centre  of  gravity  of  the  beam  being  below  the  fulcrum, 
it  may  be  made  to  approach  to  or  recede  from  it,  by  a  solid 
ball  of  metal  attached  to  the  beam  by  means  of  a  screw,  by 
which  it  may  be  raised  or  depressed  at  pleasure.  The 
remaining  conditions  of  sensibility  will  be  limited  by  the 
strength  of  the  material,  and  the  use  to  which  it  is  to  be 
applied. 

Should  it  be  found  that  a  balance  is  not  true,  it  may  still 
be  employed,  with  but  slight  error,  as  indicated  below. 

Denote  the  length  of  the  lever  arms,  by  r  and  /•',  and  the 
weight  of  the  body,  by  W.  When  the  weight  W  is  applied 
at  the  extremity  of  the  arm  r,  denote  the  counterpoising 
weights  employed,  by  W  ;  and  when  it  is  applied  at  the 
extremity  of  the  arm  /•',  denote  the  counterpoising  weights 
employed,  by  W".  We  shall  have,  from  the  principle  of  the 
lever, 

Wr  =    W'r',    and  ^Yr'  =    W'r. 

Multiplying  these  equations,  member  by  member,  we  have, 

li'V/  =    W"  li'/r';         /.      W=  y/W'  II'": 

that  is,  ill'  ti-in  weight  is  equal  to  the  squan  root  of  the  pro- 
duct of  the  apparent  weights. 

A  still  better  method,  and  one  that  is  more  free  from  the 
effects  of  errors  in  construction,  is  to  place  the  body  to  be 


Fig. 


ELEMENTARY    MACHINES.  105 

weighed  in  one  scale  and  add  counterpoising  weights  till  the 
beam  is  horizontal ;  then  remove  the  body  to  be  weighed 
and  replace  it  by  known  weights  till  the  beam  is  again  hori- 
zontal ;  the  sum  of  the  replacing  weights  will  be  the  weight 
required.  If,  in  changing  the  loads,  the  positions  of  the 
knife-edges  are  not  moved,  this  method  is  almost  exact,  but 
this  is  a  condition  difficult  to  fulfill  in  manipulation. 

The  Steelyard. 

82.     The  steelyard  is  an  instrument  used  for  weighing 
bodies.      It  consists  of  a  lever  AB,  called  the  beam ;  a  ful- 
crum E;    a    scale-pan  Z>, 
attached  at  the  extremity 
of  one  arm;  and  a  known 

weight  E,  movable    along         -aV^y^l5"Ml0' ggFft 

the  other  arm.     We  shall  /  \  E 

suppose  the  weight  of  E  to 
be  1  lb.  This  instrument 
is  sometimes  more  conve- 
nient than  the  balance,  but  it  is  more  inaccurate.  The  con- 
ditions of  sensibility  are  essentially  the  same  as  for  the 
balance.  To  graduate  the  instrument,  place  a  pound-weight 
in  the  pan  Z>,  and  move  the  counterpoise  E  till  the  beam 
rests  horizontal — let  that  point  be  marked  1 ;  next  place  a 
10  lb.  weight  in  the  pan,  and  move  the  counterpoise  E  till 
the  beam  is  again  horizontal,  and  let  that  point  be  marked 
10 ;  divide  the  intermediate  space  into  nine  equal  parts,  and 
mark  the  points  of  division  as  shown  in  the  figure.  These 
spaces  may  be  subdivided  at  pleasure,  and  the  scale  ex- 
tended to  any  desirable  limits.  We  have  supposed  that  the 
centre  of  gravity  coincides  with  the  fulcrum  ;  when  this  is 
not  the  case,  the  weight  of  the  instrument  must  be  taken 
into  account  as  a  force  applied  at  its  centre  of  gravity.  We 
may  then  graduate  the  beam  by  experiment,  or  we  may 
compute  the  lever  arms,  corresponding  to  the  different 
weights,  by  the  general  principle  of  moments. 

To  weigh  any  body  with  the  steelyard,  place  it  in  the 
scale-pan  and  move  the  counterpoise  E  along  the  beam  till 
5* 


106 


MECHANICS. 


an   equilibrium   is   established  between  the  two ;  the  cor- 
responding mark  on  the  beam  will  indicate  the  weight. 


Fig.  70. 


The  bent  Lever  Balance. 

§3.  This  balance  consists  of  a  bent  lever  ACB 
fulcrum  C ;  a  scale-pan 
D ;  and  a  graduated  arc 
EF,  whose  centre  co- 
incides with  the  centre 
of  motion  C.  When  a 
weight  is  placed  in  the 
scale-pan,  the  pan  is  de- 
pressed and  the  lever- 
arm  of  the  weight  is 
diminished  ;  the  weight  B  is  raised,  and  its  lever-arm 
increased.  When  the  moments  of  the  two  forces  become 
equal,  the  instrument  will  come  to  a  state  of  rest,  and  the 
weight  will  be  indicated  by  a  needle  projecting  from  _Z>,  and 
playing  in  front  of  the  arc  FE.  The  zero  of  the  arc  EF  is 
at  the  point  indicated  by  the  needle  when  there  is  no  load  in 
the  pan  D. 

The  instrument  may  be  graduated  experimentally  by 
placing  weights  of  1,  2,  3,  &c,  pounds  in  the  pan,  and  mark- 
ing the  points  at  which  the  needle  comes  to  rest,  or  it  may 
be  graduated  by  means  of  the  general  principle  of  moments. 
We  need  not  explain  this  method  of  graduation. 

To  weigh  a  body  with  the  bent  lever  balance,  place  it  in 
the  scale-pan,  and  note  the  point  at  which  the  needle  comes 
to  rest ;  the  reading  will  make  known  the  weight  sought. 


Compound  Balances. 

84.  Compound  balances  are  much  used  in  weighing 
heavy  articles,  as  merchandise,  coal,  freight  for  shipping, 
&c.  A  great  variety  of  combinations  have  been  employed, 
one  of  which  is  annexed. 

A  B  is  a  platform,  on  which  the  object  to  be  weighed  is 


ELEMENTARY    MACHINES. 


107 


Fig.  71. 


placed  ;  B  C  is   a  guard 
firmly   attached    to    the 

platform ;     the    platform 

is    supported    upon     the 

knife-edge     fulcrum     F, 

and  the  piece  D,  through 

the   medium   of  a  brace 

CD  ;   GF'\§  a  lever  turn- 

ing  about  the  fulcrum  F, 

and  suspended  by  a  rod  from  the  point  L  ;  LN  is  a  lever 

having  its  fulcrum  at  M,  and  sustaining  the  piece  D  by  a 

rod  KH\  0  is  a  scale-pan  suspended  from  the  end  N  oi  the 

lever  LN.     The  instrument  is  so  constructed,  that 

BF\   GF::  KM:  LM; 

and  the  distance  JTM  is  generally  made  equal  to  y1^  of  3IN. 
The  parts  are  so  arranged  that  the  beam  LN  shall  rest 
horizontally  in  equilibrium  when  no  weight  is  placed  on  the 
platform. 

If,  now,  a  body  Q  be  placed  upon  the  platform,  a  part  of 
its  weight  will  be  thrown  upon  the  piece  X>,  and,  acting 
downwards,  will  produce  an  equal  pressure  at  K.  The 
remaining  part  will  be  thrown  upon  F,  and,  acting  upon  the 
lever  FG,  will  produce  a  downward  pressure  at  67,  which 
will  be  transmitted  to  L ;  but,  on  account  of  the  relation 
given  by  the  above  proportion,  the  effect  of  this  pressure 
upon  the  lever  Xi\rwill  be  the  same  as  though  the  pressure 
thrown  upon  E  had  been  applied  directly  at  K.  The  final 
effect  is,  therefore,  the  same  as  though  the  weight  of  Q  had 
been  applied  at  7T,  and,  to  counterbalance  it,  a  weight  equal 
to  T\  of  Q  must  be  placed  in  the  scale-pan  0. 

To  weigh  a  body,  then,  by  means  of  this  scale,  place  it  on 
the  platform,  and  add  weights  to  the  scale-pan  till  the  lever 
LN  is  horizontal,  then  10  times  the  sum  of  the  weight 
added  will  be  equal  to  the  weight  required.  By  making 
other  combinations  of  levers,  or  by  combining  the  princi- 


108  MECHANICS. 

pie   of  the   steelyard   with   this   balance,   objects   may  be 
weighed  by  using  a  constant  counterpoise. 

EXAMPLES. 

1.  In  a  lever  of  the  first  class,  the  lever  arm  of  the 
resistance  is  2|  inches,  that  of  the  power,  334n  and  the 
resistance  100  lbs.  What  is  the  power  necessary  to  hold 
the  resistance  in  equilibrium  ?  Aus.  8  lbs. 

2.  Four  weights  of  1,  3,  5,  and  7  lbs.  respectively,  are 
suspended  from  points  of  a  straight  lever,  eight  inches  apart. 
How  far  from  the  point  of  application  of  the  first  weight 
must  the  fulcrum  be  situated,  that  the  weights  may  be  in 
equilibrium  ? 

SOLUTION. 

Let  x  denote  the  required  distance.    Then,  from  Art.  (36) 
1  X  x  +  3(x  —  8)  +  5(x  -  16)  +  7(flJ  -  24)    =   0  ; 
/.     x  —   17  in.     A?is. 

3.  A  lever,  of  uniform  thickness,  and  12  feet  long,  is 
kept  horizontal  by  a  weight  of  100  lbs.  applied  at  one 
extremity,  and  a  force  P  applied  at  the  other  extremity,  so 
as  to  make  an  angle  of  30°  with  the  horizon.  The  fulcrum 
is  20  inches  from  the  point  of  application  of  the  weight,  and 
the  weight  of  the  lever  is  10  lbs.  What  is  the  value  of  P, 
and  what  is  the  pressure  upon  the  fulcrum  ? 

SOLUTION. 

The  lever  arm  of  P  is  equal  to  124  in.  x  sin  30°  =  62  in., 
and   the  lever   arm   of  the  weight   of  the  lever  is  52  in. 
Hence, 
20  x  100  =  10  x  52  +  P  x  62  ;        .-.    P  =  24  lbs.  nearly 

We  have,  also, 


R   =    x/X2  +  Y1  =  VXllO  +  24  Sd  30°)2  "r  (24cos30°)'. 
.*.     B  =  123.8  lbs.  ; 


ELEMENTABT    MACHINES.  109 

X         20.785  __ 

and,  cos  a   =   -g  =   ^-g-  =  .16789; 

.-.     a  —   80°  20'  02". 

4.  A  .leavy  lever  rests  on  a  fulcrum  which  is  2  feet  from 
one  end,  8  feet  from  the  other,  and  is  kept  horizontal  by  a 
weight  of  100  lbs.,  applied  at  the  first  end,  and  a  weight 
of  18  lbs.,  applied  at  the  other  end.  What  is  the  weight 
of  the  lever,  supposed  of  uniform  thickness  throughout  ? 

SOLUTION. 

Denote  the  required  weight  by  x ;  its  arm  of  lever  is 
3  feet.     We  have,  from  the  principle  of  the  lever, 

100  X  2  :=  x  x  3  +  18  X  8  ;         .\     x  =  18f  lbs.   Ans. 

5.  Two  weights  keep  a  horizontal  lever  at  rest ;  the 
pressure  on  the  fulcrum  is  10  lbs.,  the  difference  of  the 
weights  is  4  lbs.,  and  the  difference  of  lever  arms  is  9  inches. 
What  are  the  weights,  and  their  lever  arms  ? 

Ans.  The  weights  are  7  lbs.  and  3  lbs. ;  their  lever  arms 
are  15|  in.,  and  6f  in. 

6.  The  apparent  weight  of  a  body  weighed  in  one  pan 
of  a  false  balance  is  5j  lbs.,  and  in  the  other  pan  it  is 
6T6T  lbs.     What  is  the  true  weight  ? 


TF=  yy    x  ff  =  6  lbs.  Ans. 

7.     In  the  preceding  example,  what  is  the  ratio  of  the 
lever  arms  of  the  balance  ? 

SOLUTION. 

Denote  the  shorter  arm  by  I,  and  the  longer  arm  by  »/, 
We  shall  have,  from  the  principle  of  moments, 

6/  =  5i  x  nl,   or,  6?il  =  QT6Tl ;         .-.     n  =  \T\. 

That  is,  the  longer  arm  equals  1TTT  times  the  shorter  arm. 


ilO  MKCHANICS. 


The  Inclined  Plane. 


85.  An  inclined  plane  is  a  plane  inclined  to  the  horizon. 
In  this  machine,  let  the  power  be  a  force  applied  to  a  body 
either  to  prevent  motion  down  the  plane,  or  to  produce 
motion  np  the  plane,  and  let  the  resistance  be  the  weight  of 
the  body  acting  vertically  downwards.  The  power  may  be 
applied  in  any  direction  whatever ;  but  we  shall,  for  sim- 
plicity's sake,  suppose  it  to  be  in  a  vertical  plane,  taken  per- 
pendicular to  the  inclined  plane. 

Let  AB  represent  the  inclined  plane,  0  a  body  resting 
on  it,  R  the  weight  of  the  body, 
and  P  the  force  applied  to  hold  it 
in  equilibrium.  In  order  that  these 
two  forces  may  keep  the  body  at 
rest,  friction  being  neglected,  their 
resultant  must  be  perpendicular  to 
AB  (Art.  72).  _  Fig72. 

When  the  direction  of  the  force 
P  is  given,  its  intensity  may  be  found  geometrically,  as  fol- 
lows :  draw  OR  to  represent  the  weight,  and  0  Q  perpen- 
dicular to  AB ;  through  R  draw  RQ  parallel  to  OP,  and 
through  Q  draw  QP  parallel  to  OR ;  then  will  OP  repre- 
sent the  required  intensity,  and  OQ  the  pressure  on  the 
plane. 

When  the  intensity  of  P  is  given,  its  direction  may  be 
found  as  follows :  draw  OR  and  OQ  as  before  ;  with  R  as 
a  centre,  and  the  given  intensity  as  a  radius,  describe  an 
arc  cutting  OQ'm  Q;  draw  RQ,  and  through  0  draw  OP 
parallel,  and  equal  to  R  Q ;  it  will  represent  the  direction 
of  the  force  P. 

If  we  denote  the  angle  between  P  and  R  by  <p,  aod  the 
inclination  of  the  plane  by  «,  we  shall  have  the  angle  ROQ 
equal  to  a,  since  OQ  is  perpendicular  to  AB,  and  OR  to 
A  C\  and,  consequently,  the  angle  QOP  =  cp  ~  a,  From 
the  principle  of  Art.  35,  we  have, 

P  :  R  :  :  sin«  :  sin(<p  -«)    .     .     (  38.) 


ELEMENTARY    MACHINES. 


Ill 


From  which,  if  either  P  or  p  be  given,  the  other  can  be 
found. 

If  we  suppose  the  power  to  be 
applied  parallel  to  the  plane,  we 
shall  have,  <p  —  a    =  90°, 

or,  sin(p  —  a)  =     1. 

We  have,  also,         sin  a    =  —--  • 

AH 

Substituting  these  in  the  preced- 
ing proportion,  and  reducing,  we 
have,  • 


Fig.  73. 


P :  B  :  :  BO  :  AB 


(39.) 


That  is,  when  the  power  is  parallel  to  the  plane,  the  power 
is  to  the  resistance,  as  the  height  of  the  plane  is  to  its  length. 

If  the  power  is  parallel  to  the  base  of  the  plane,  we  shall 
have,  (p  —  a   =  90°  —  a ;  whence, 


also, 


sin  (s  —  a)  =   cos  a 
BO 


AG 
AB; 


sm  a   = 


AB 


Substituting  in  Proportion   (38), 
and  reducing,  we  have, 

JP  :  B  :  :  BG 


Fig.  74. 


AC 


(40.) 


That  is,  the  power  is  to  the  resistance  as  the  height  of  the 
plane  is  to  its  base. 

From  the  last  proportion  we  have, 


JiAC 


Bt&nct . 


If  we  suppose  a  to  increase,  the  value  of  P  will  increase, 
and  when  a  becomes  90°,  P  will  become  infinite  ;  that  is,  if 
friction  be  neglected,  no  finite  horizontal  force  can  sustain  a 
body  against  a  vertical  wall. 


112  MECHANICS. 

EXAMPLES. 

1.  A  power  of  1  lb.,  acting  parallel  to  an  inclined  plane, 
supports  a  weight  of  2  lbs.  What  is  the  inclination  of  the 
plane  ?  Ans.  30°. 

2.  The  power,  resistance,  and  normal  pressure,  in  the 
case  of  an  inclined  plane,  are,  respectively,  9,  13,  and  6  lbs. 
AVI i at  is  the  inclination  of  the  plane,  and  what  angle  docs 
the  power  make  with  the  plane  ? 

SOLUTION. 

If  we  denote  the  angle  between  the  power  and  resistance 
by  <p,  and  the  inclination  of  the  plane  by  a,  we  shall  have, 
from  Art.  (35), 

6    =    V'132  +  92  +  2  X  9  X  13  cos  9; 
/.     9   =:   156°  8'  20". 
Also,  from  Art.  (35),  for  the  inclination  of  the  plane, 
6:9::  sin  156°  8'  20"  :  sin  a  ;         .-.     a   =   37°  21'  26". 

Inclination  of  power  to  plane  =  9  —  90°  —  a  =  28°  46'  54". 

Arts. 

3.  A  body  may  be  supported  on  an  inclined  plane  by  a 
force  of  10  lbs.,  acting  parallel  to  the  plane  ;  but  it  requires 
a  force  of  12  lbs.  to  support  it  when  the  force  acts  parallel 
to  the  base.  What  is  the  weight  of  the  body,  and  what  is 
the  inclination  of  the  plane  ? 

Ans.  The  weight  is  18.09  lbs.,  and  the  inclination  is 
33°  33'  25". 

The  Pulley. 

^.<i.     A  pulley  consists  of  a  wheel  having  a  groove  around 
-  circumference  to  receive  a  cord;  the  wheel  turns  freely 
•  n  an  axis  at  righl   angles  to  its  plane,  which  axis  is  sup- 
ported by  a  frame  called  a  block.    The  pulley  is  said  to  be 

fixed,  when  the  block  is  fixed,  and  to  be  movable,   when 


ELEMENTARY    MACHINES. 


113 


the  block  is  movable.     Pulleys  may  be  used  singly,  or  in 

combinations. 

Single  fixed  Pulley. 

87.  In  this  pulley  the  block,  and,  consequently,  the  axis, 
is  fixed.  Denote  the  power  by  1\  the  resist- 
ance by  B,  and  the  radius  of  the  pulley  by  r. 
It  is  plain  that  both  the  power  and  resistance 
should  be  in  a  plane,  at  right  angles  to  the 
axis.  Hence,  if  we  take  the  axis  of  the  pulley 
as  the  axis  of  moments,  we  shall  have  (Art.  49), 
the  following  condition  of  equilibrium : 


Pr  =  Br ;    or,  P  =  P. 


foh 


P 

Fig. 


R 


That  is,  in  the  single  fixed  pulley,  the  power  is  equal  to  the 
resistance. 

The  effect  of  the  pulley  is,  therefore,  simply  to  change  the 
direction  of  the  force,  and  it  is  for  this  purpose  that  it  is 
generally  used. 

Single  Movable  Pulley. 

88.  In  this  pulley  the  block,  and,  consequently,  the 
axis,  is  movable.  The  resistance  is  applied  at 
a  hook  attached  to  the  block ;  one  end  of  a 
rope,  enveloping  the  lower  part  of  the  pulley, 
is  firmly  attached  at  a  fixed  point  (7,  and  the 
power  is  applied  at  the  other  extremity.  We 
shall  take  the  two  branches  of  the  rope  par- 
allel, that  being  the  most  advantageous  way  of 
using  the  machine. 

Adopting  the  notation  of  the  preceding 
article,  and  taking  A,  the  point  of  contact  of 
CA  with  the  pulley,  as  the  centre  of  moments, 
we  shall  have,  for  the  condition  of  equilibrium 
(Art.  49), 

P  x  %r  =  Pr;        .:     P  =  ±B. 

That  is,  in  the  movable  pulley,  when  the  power  and 
resistance  are  parallel,  the  poicer  is  equal  to  one  half  of  the 
resistance.     The  tension  upon  the  cord  CA  is  evidently  the 


114 


MECHANICS. 


same  as  that  upon  the  cord  BP.  It  is,  therefore,  equal  to 
the  power,  or  to  one-half  the  resistance.  If,  therefore, 
the  resistance  of  the  fixed  point  C  be  replaced  by  a  force 
equal  to  P,  the  equilibrium  will  be  undisturbed. 

If  the  two  branches  of  the  en- 
veloping cord  are  oblique  to  each 
other,  the  condition  of  equilibrium 
will  be  somewhat  modified.  Sup- 
pose the  resistance  of  the  fixed 
point  C  to  be  replaced  by  a  force 
equal  to  P,  and  denote  the  angle 
between  the  two  branches  of  the 
cord  by  2 p.  If  an  equilibrium 
subsists  between  the  forces  P,  P, 
and  P,  we  must  have  the  relation, 


2  Pcos?  =  B. 

Draw  the  chord  AB  between  the  points  of  contact  of  the 
cord  and  pulley,  and  denote  its  length  by  c ;  draw,  also, 
the  radius  OB.  Then,  since  OB  is  perpendicular  to  AB 
and  BP  to  OP,  the  angle  ABO  will  be  equal  to  one  half 
of  the  angle  ACB,  or  equal  to  <p.     Hence, 


COS(p 


ic  -f-  r 


c 
2r 


Substituting  in  the  preceding  equation  and  reducing,  we 
have, 

Pc  =  Br;         .'.     P  :  B     :  r  :  c    .     .     (41.) 

That  is,  the  power  is  to  the  resistance  as  the  radius  of  the 
pulley  is  to  the  chord  of  the  arc  enveloped  by  the  rope. 

When  the  chord  is  greater  than  the  radius,  there  will 
be  a  gain  of  mechanical  advantage  in  the  use  of  this  pulley; 
when  less,  there  will  be  a  loss  of  mechanical  advantage. 

If  the  chord  becomes  equal  to  the  diameter,  we  have,  as 

before, 

P  =-  IB. 


ELEMENTARY    MACHINES. 


115 


Combinations  of  Separate  Movable  Pulleys. 

89.  The  figure  represents  a  combination  of  three  movable 
pulleys,  in  which  there  are  as  many  separate 
cords  as  there  are  pulleys ;  the  first  end  of 
each  cord  is  attached  at  a  fixed  point,  the 
second  end  being  fastened  to  the  hook  of 
the  next  pulley  in  order,  except  the  last 
cord,  at  the  second  extremity  of  which  the 
power  is  applied. 

Let  us  denote  the  tension  of  the  cord 
between  the  first  and  second  pulley  by  £, 
that  of  the  cord  between  the  second  and 
third  pulley  by  t'.  By  the  preceding 
Article,  Ave  have, 


iiJ: 


*' 


¥; 


¥'■ 


Multiplying  these  equations  together,  member  by  member, 
and  striking  out  the  common  factors  in  the  resulting  equa- 
tion, we  have, 

P  =  (|)3i£. 

Had  there  been  n  pulleys  in  the  combination,  we  should 
have  obtained,  in  an  entirely  similar  manner,  the  relation, 


P  =    (D-.22; 


P:  R 


42.) 


That  is,  the  power  is  to  the  resistance  as  1  is  to  2",  n 
denoting  the  number  of  pulleys. 

For  convenience,  the  last  branch  of  the  cord  is  often 
passed  over  a  fixed  pulley  ;  this  arrangement  only  serves  to 
change  the  direction  of  the  force,  without  in  any  way  chang- 
ing the  conditions  of  equilibrium. 

Combinations  of  Pulleys  in  blocks. 

90.     These   combinations   are   effected  in  a  variety  of 

ways.     In  most  cases,  there  is  but  a  single  rope  employed, 

which,  being  firmly  attached  to  a  hook  of  one  block,  passes 

around  a  pulley  in  the  other  block,  then  around  one  in  the 


116 


MECHANICS. 


first  block,  and  so  on,  passing  from  block  to  block  until  it 
has  passed  around  each  pulley  in  the  system.  The  power  is 
applied  at  the  free  end  of  the  rope.  Sometimes  the  pulleys 
in  the  same  block  are  placed  side  by  side,  sometimes  they 
are  placed  one  above  another,  as  represented  in  the  figure, 
in  which  case  the  interior  pullies  are  made 
somewhat  smaller  than  the  outer  ones.  The 
conditions  of  equilibrium  are  the  same  in  both 
cases.  To  deduce  the  conditions  of  equili- 
brium in  the  case  represented,  in  which  the 
upper  block  is  fixed  and  the  lower  one  mov- 
able :  denote  the  power  by  I\  the  resistance 
by  R.  When  there  is  an  equilibrium  between 
P  and  it,  the  tension  upon  each  branch  of 
the  rope  which  aids  in  supporting  the  resist- 
ance must  be  the  same,  and  equal  to  P;  but, 
since  the  last  pulley  simply  serves  to  change 
the  direction  of  the  force  P,  there  will  be 
four  such  branches  in  the  case  considered ; 
hence,  we  shall  have, 


4P  =  i?,     or 


P  =ix. 


Had  there  been  n  pulleys  in  the  combination,  there  would 
have  been  n  supporting  branches  of  the  cord,  and  we  should 
have  had,  in  the  same  maimer, 

•      (43.) 


nP  =  72,    or  P  :  R 


1  :  n 


That  is,  the  power  is  to  the  resistance  as  1  is  to  the  num- 
ber of  branches  of  the  rope  which  support  the  resistance. 

The  principles  involved  in  the  combinations  already  con- 
sidered, will  be  sufficient  to  make  known  the  relation 
between  the  power  and  resistance  in  any  combination  what- 
ever. 

EZiXPL  i:  s  . 

1.  In  a  Bystem  of  six  movable  pulleys,  of  the  kind  des- 
cribed in  Art.  89,  what  weight  can  be  sustained  by  a  \ tower 
ofl2jbs?  Ans.  768  lbs. 


ELEMENTARY    MACHINES. 


117 


2.  In  a  combination  of  pulleys  in  two  blocks,  when 
there  are  six  pulleys  in  each  block,  what  weight  can  a  power 
of  12  lbs.  sustain  in  equilibrium?  Ans.  144  lbs. 

3.  In  a  combination  of  separate  movable  pulleys,  the 
resistance  is  576  lbs.,  and  the  power  which  keeps  it  in  equi- 
librium is  9  lbs.  How  many  pulleys  are  there  in  the  com- 
bination ?  Ans.  6. 

4.  In  a  combination  of  pulleys  in  two  blocks,  with  a  single 
rope,  the  power  is  62  lbs.,  and  the  resistance  496  lbs.  How 
many  pulleys  are  there  in  each  block  ?  Ans.  4. 

5.  In  a  combination  of  two  movable  pulleys,  the  inclina- 
tions of  the  ropes  at  each  pulley  is  120°.  What  is  the  power 
required  to  support  a  weight  of  27  lbs.  ?  Ans.  9  lbs. 

The  Wheel  and  Axle. 

91.  The  wheel  and  axle  consists  of  a  wheel  A,  mounted 
on  an  axle  or  arbor  B.  The  power  is 
applied  at  one  extremity  of  a  rope 
wrapped  around  the  wheel,  and  the 
resistance  at  one  extremity  of  a  sec- 
ond rope,  wrapped  around  the  axle  in 
a  contrary  direction.  The  whole  in- 
strument is  supported  by  pivots  pro- 
jecting from  the  ends  of  the  axle.  Iu 
deducing  the  conditions  of  equili- 
brium of  the  power  and  resistance,  we  shall  suppose  them  tc 
be  situated  in  planes,  at  right  angles  to  the  axis. 

Denote  the  power  by  P,  the  re- 
sistance by  P,  the  radius  of  the 
wheel  by  r,  and  the  radius  of  the 
axle  by  r'.  We  shall  have,  in  case 
of  an  equilibrium  (Art.  49), 


F-g   30. 


Pr=Rr\  orP  :  P 


(44.) 


That  is,  the  power  is  to  the  resistance 
as  the  radius  of  the  axle  is  to  the 
radius  of  the  wheel. 


ns 


MECHANICS. 


By  suitably  varying  the  dimensions  of  the  wheel  and  axle, 
any  amount  of  mechanical  advantage  may  be  obtained. 

II'  we  draw  a  straight  line  from  the  point  of  contact  of 
the  first  rope  and  the  wheel,  to  the  point  of  contact  of  the 
second  rope  and  the  axle,  the  [tower  and  resistance  being 
parallel,  it  can  readily  be  shown  that  it  will  cut  the  axis  of 
revolution  at  a  point  which  divides  the  line  through  the 
points  of  contact  into  two  parts,  which  are  inversely  pro- 
portional to  the  power  and  resistance.  Hence,  this  is  the 
point  of  application  of  the  resultant  of  these  two  forces. 
The  resultant  will  be  equal  to  the  sum  of  the  forces,  and  by 
the  aid  of  the  principle  of  moments,  the  pressure  on  each 
pivot  maybe  computed.  When  the  weight  of  the  machine 
is  to  be  taken  into  account,  we  must  regard  it  as  a  vertical 
force  applied  at  the  centre  of  gravity  of  the  wheel  and  axle. 
The  pressures  upon  each  pivot  due  to  this  weight,  may  be 
computed  separately,  and  added  to  those  already  found. 


Combinations  of  Wheels  and  Axles. 

92.  If  the  rope  of  the  first  axle  be  passed  around  a 
second  wheel,  and  the  rope  of  the  second  axle  around  a 
third  wheel,  and  so  on,  a  combination  will  result  which  is 
capable  of  affording  great  mechanical  advantage.  The 
figure  represents  a  combination  of  two 
Avheels  and  axles.  To  deduce  the 
conditions  of  equilibrium,  denote  the 
power  by  P,  the  resistance  by  J?,  the 
radius  of  the  first  wheel  by  r,  that  of 
the  first  axle  by  r\  that  of  the  second 
wheel  by  r",  and  that  of  the  second 
axle  by  r'".  If  we  denote  the  tension 
of  the  connecting  rope  by  f,  this  may 
be  regarded  as  a  power  applied  to  the 
second  wheel.  From  what  was  de- 
monstrated for  the  wheel  and  axle,  we 
shall  have, 

Pr  =  tr\    and  tr"  =  Br"\ 


ELEMENTARY    MACHINES. 


119 


Multiplying  these  equations  together,  member  by  member, 
and  reducing,  we  have, 

Prr"  f=  Br'r'"  ;    or,  P  :  B 


:  rr 


In  like  manner,  were  there  any  number  of  Avheels  and 
axles  in  the  combination,  we  might  deduce  the  relation, 

Prr"r"  .  .  .   =  Br'r'"rv 


or, 


P  :  B  :: 


rr  r 


(45.) 


That  is,  the  power  is  to  the  resistance  as  the  continued 
product  of  the  radii  of  the  axles  is  to  the  continued  product 
of  the  radii  of  the  wheels. 

The  principle  just  explained,  is  applicable  to  those  kinds 
of  machinery  in  which  motion  is  transmitted  from  wheel  to 
wheel  by  the  aid  of  bands,  or  belts.  An  endless  band, 
called  the  driving  belt,  passes  around  one  drum  mounted 
upon  the  axle  of  the  driving  wheel,  and  around  another  on 
that  of  the  driven  wheel.  When  the  radius  of  the  former 
is  greater  than  that  of  the  latter,  there  is  a  gain  of  velocity, 
and  a  corresponding  loss  of  power ;  in  the  contrary  case, 
there  is  a  loss  of  velocity,  and  a  corresponding  gain  of 
power.  In  the  first  case,  we  are  said  to  gear  up  for  velo- 
city ;  in  the  second  case,  we  are  said  to  gear  down  for 
power.  These  remarks  admit  of  extension  to  combinations 
of  any  number  of  pieces,  in  which  motion  is  transmitted  bj 
belts,  cords,  chains,  or,  as  we  shall  see  hereafter,  by  trains 
of  toothed  wheels. 


93, 


The  Crank  and  Axle,  or  Windlass. 
This  machine  con- 


sists of  an  axle  AB,  and  a 
crank  B  CD.  The  power. 
is  applied  to  the  crank-han- 
dle D  C,  and  the  resistance 
to  a  rope  wrapped  around 
the  axle.  The  distance  from 
the  handle  DC  to  the  axis, 
is  called  the  crank-arm. 


Fig.  88. 


120 


MECHANICS. 


The  relation  between  the  power  and  resistance,  when  in 
equilibrium,  is  the  same  as  in  the  wheel  and  axle,  except 
that  we  substitute  the  crank-arm  for  the  radiu^  of  the  wheel. 

Hence,  the  power  is  to  the  resistance  as  the  radius  of  the 
axle  is  to  the  crank-arm. 

This  machine  is  used  in  drawing  water  from  wells,  raising 
ore  from  mines,  and  the  like.  It  is  also  used  in  combina- 
tion with  other  machines.  Instead  of  the  crank,  as  shown 
in  the  figure,  two  holes  are  sometimes  bored  at  right  angles 
to  each  other  and  to  the  axis,  and  levers  inserted,  at  the  ex- 
tremities of  which  the  power  is  applied.  The  condition  of 
equilibrium  remains  unchanged,  provided  we  substitute  for 
the  crank-arm,  the  distance  from  the  point  of  application  of 
the  power  to  the  axis. 

The  Capstan. 

94.  The  Capstan  differs  in  no  material  respect  from  the 
windlass,  except  in  having  its  axis  vertical.  The  capstan 
consists  of  a  vertical  axle  passing  through  strong  guides, 
and  having  holes  at  its  upper  end  for  the  insertion  of  levers. 
It  is  much  used  on  shipboard  for  raising  anchors.  The  con- 
ditions of  equilibrium  are  the  same  as  in  the  windlass. 

The  Differential  Windlass. 

95.  This  differs  from  the  common  windlass  in  having  an 
axle  formed  of  two  cylinders, 

A   and  B.   of  different   dia-        _ 
meters,   but   having   a    com- 
mon axis.    A  rope  is  attached 
to  the    larger  cylinder,    and 

wrapped  several  times  around  d  ■  m» 

it,  after  which  it  passes  around  t,l       It— iB. 

the  movable  pulley  C,  and, 
returning,  is  wrapped  in  a 
contrary  direction  about  the 
smaller  cylinder,  to  which  the 
second  end  of  the  rope  is 
made  fast.     The  power  is  ap- 


■E 


Fig.  84. 


plied  at  the  crank-handle  FE,  and  the  resistance  to  the  hook 
of  the  movable  pulley.     When  the  crank  is  turned  so  as  to 


ELEMENTAET    MACHINK8. 


121 


wind  the  rope  upon  the  larger  cylinder  it  unwinds  from 
the  smaller  one,  but  in  a  less  degree,  and  the  total  effect  of 
the  power  is  to  raise  the  resistance  JR.  To  deduce  the 
conditions  of  equilibrium  between  the  power  and  resistance, 
denote  the  power  by  1\  the  resistance  by  li,  the  crank-arm 
by  c,  the  radius  of  the  larger  cylinder  by  ;•,  and  that  of  the 
smaller  cylinder  by  r'.  The  resistance  acts  equally  upon 
the  two  branches  of  the  rope  from  which  it  is  suspended, 
hence  the  tension  of  each  branch  may  be  represented  by 
±M.  Suppose  that  the  power  acts  to  wind  the  rope  upon 
the  larger  cylinder.  The  moment  of  the  power  will  be 
Pc ;  the  moment  of  the  tension  of  the  branch  A  will  be  equal 
to  \Rr\  this  acts  to  assist  the  power ;  the  moment  of  the 
tension  of  the  branch  B  will  be  equal  to  \Rr,  this  acts  to  op- 
pose the  power.     From  the  principle  of  moments,  we  have, 


whence, 


Pc  +  \Rr'  =  \Pi\   or  Pc  =  \B  {r  -  r')  ; 

P  :  R  :  :  r  -  r'  :  2c.     .     .     .     ( 46.) 


That  is,  the  power  is  to  the  resistance  as  the  difference  of 
the  radii  of  the  two  cylinders  is  to  twice  the  crank-arm. 

By  increasing  the  crank-arm  and  diminishing  the  differ- 
ence between  the  radii  of  the  cylinders,  any  amount  of 
mechanical  advantage  may  be  obtained  by  the  use  of  this 
machine. 

Wheel-work. 


96.  The  principle  employed  in  findin 
between  the  power  and  resist- 
ance in  a  train  of  wheel-work 
is  the  same  as  that  used  in 
discussing  the  wheel  and  axle 
and  its  modifications.  To  illus- 
trate the  method  of  proceed- 
ing, we  have  taken  the  case  in 
which  the  power  is  applied  to  a 
crank-handle  which  is  attached 
to  the  axis  of  a  cogged  wheel 
6 


the    relation 


Fig.  85 


122 


MECHANICS. 


Fig.  85. 


A ;  the  teeth,  or  cogs,  of  this  wheel  work  into  the  spaces 
of  the  toothed  wheel  B,  and 
the  resistance  is  attached  to  a 
rope  wound  round  the  arbor 
of  the  last  wheel.  In  order 
that  the  wheel  A  may  com- 
municate motion  freely  to  the 
wheel  B,  the  number  of  teeth 
in  their  circumferences  should 
be  proportional  to  their  radii, 
and  the  spaces  between  the 
teeth  in  one  wheel  should  be  large  enough  to  receive  the 
teeth  of  the  other  wheel,  but  not  large  enough  to  allow  a 
great  deal  of  play.  The  teeth  should  always  come  in  contact 
at  the  same  distances  from  the  centres  of  the  wheels,  and 
those  distances  are  taken  as  the  radii  of  the  wheels  them- 
selves. Denote  the  power  by  B,  the  resistance  by  B,  the 
crank-arm  by  c,  the  radius  of  the  wheel  A  by  r,  that  of  the 
wheel  B  by  r',  that  of  the  arbor  by  r",  and  suppose  the 
power  and  resistance  to  be  in  equilibrium  ;  then  will  the 
pressure  due  to  the  action  of  the  power  tend  to  turn  the 
wheels  in  the  direction  of  the  arrow  heads.  This  tendency 
will  be  counteracted  by  the  pressure  of  the  resistance  tend- 
ing to  produce  motion  in  a  contrary  direction.  If  we 
denote  the  pressure  at  the  point  0  by  B\  we  should  have, 
from  what  has  preceded, 

Be  =  B'r    and   B'r'  =  Br" ; 


whence,  by  multiplication  and  reduction, 

Bcrr  =  Brr",     or    B  :  B  :  :  rr"  :  cr'     .     ( 47.) 

That  is,  the  power  is  to  the  resistance  as  the  continued 
product  of  the  alternate  arms  of  lever,  beginning  at  the 
resistance,  is  to  the  coyitinued  product  of  the  alternate  arms 
of  lever  beginning  at  the  power. 

Had  there  been  any  number  of  wheels  in  the  train  lying 


ELEMENTARY    MACHINES.  123 

between  the  power  and  resistance,  we  should   have  found 
oiinilar  conditions  of  equilibrium. 

EXAMPLES  . 

1.  A  power  of  5  lbs.,  acting  at  the  circumference  of  a 
wheel  whose  radius  is  5  feet,  supports  a  resistance  of  200 lbs. 
applied  at  the  circumference  of  the  axle.  What  is  the 
radius  of  the  axle  ?  A?is.  H  inches. 

2.  The  radius  of  the  axle  of  a  windlass  is  3  inches,  and 
the  crank-arm  15  inches.  What  power  must  be  applied  to 
the  crank-handle,  to  support  a  resistance  of  180  lbs.,  applied 
to  the  circumference  of  the  axle  ?  Ans.  36  lbs. 

3.  A  power  Py  acts  upon  a  rope  2  inches  in  diameter, 

passing  over  a  wheel  whose  radius  is  3  feet,  and  supports  a 

resistance  of  320  lbs.,  applied  by  a  rope  of  the  same  diame 

ter,  passing  over  an  axle  whose  radius  is  4  inches.     What  is 

the  value  of  P,  when  the  thickness  of  the  rope  is  taken  into 

account.  Ans.  43^  lbs. 

The  Screw. 

97.  The  screw  is  essentially  a  combination  of  two  in 
clined  planes.  It  consists  of  a  solid  cylinder, 
called  the  cylinder  of  the  screw,  which  is  en- 
veloped by  a  spiral  projection  called  the 
thread.  The  thread  may  be  generated  as 
follows  :  let  an  isosceles  triangle  be  placed  so 
that  its  base  shall  coincide  with  an  element  of 
the  cylinder  of  the  screw,  and  so  that  its 
plane  shall  pass  through  the  axis.     Let  the  Fig.  86. 

triangle  be  revolved  uniformly  about  the  axis, 
and  at  the  same  time  be  moved  uniformly  in  the  direction 
of  the  axis,  at  such  a  rate  that  it  shall  pass  over  a  distance 
in  this  direction  equal  to  the  base  of  the  triangle  during  one 
revolution.  The  solid  generated ^  by  the  triangle  is  the 
thread  of  the  screw.  The  two  sides  of  the  triangle  generate 
helicoidal  surfaces,  which  constitute  the  upper  and  lower 
surfaces  of  the  thread.  Every  point  in  these  lines  generates 
a  curve  called  a  helix,  which  is  entirely  similar  to  an  inclined 


124:  MECHANICS. 

plane  bent  around  a  cylinder.  The  vertex  generates  what 
is  called  the  outer  helix,  and  the  two  angular  points  of  the 
base  trace  out  the  same  curve,  which  is  the  inner  helix. 
The  screw  just  described  is  called  a  screw  with  a  triangular 
thread.  Had  we  used  a  rectangle,  instead  of  a  triangle,  and 
imposed  the  condition,  that  the  motion  in  the  direction  of 
the  axis  during  one  revolution,  should  be  equal  to  twice  the 
base,  we  should  have  had  a  screw  with  a  rectangular  thread, 
as  in  the  figure. 

The  screw  works  into  a  piece  called  a  nut,  which  is  gene- 
rated in  a  manner  entirely  analogous  to  that  just  described, 
except  that  what  is  solid  in  the  screw  is  wanting  in  the  nut  ; 
it  is,  therefore,  exactly  adapted  to  receive  the  thread  of  the 
screw.  Sometimes,  the  screw  remains  fast,  and  the  nut  is 
turned  upon  it ;  in  which  case,  the  nut  has  a  motion  of  revo- 
lution, combined  with  a  longitudinal  motion.  Sometimes, 
the  nut  remains  fast,  and  the  screw  is  turned  within  it,  in 
which  case,  the  screw  receives  a  motion  in  the  direction  of 
its  axis,  in  connection  with  a  motion  of  rotation.  The  con- 
ditions of  equilibrium  are  the  same  for  each.  In  both  cases, 
the  power  is  applied  at  the  extremity  of  a  lever;  when  in 
motion,  the  point  of  application  describes  an  ascending  or 
descending  spiral,  resulting  from  a  combination  of  the 
rotary  and  the  longitudinal  motion  We  shall  suppose  the 
nut  to  remain  fist,  and  the  screw  to  be  movable,  and  that 
the  resistance  acts  parallel  to  the  axis  of  the  screw.  If  the 
axis  is  vertical,  and  the  resistance  a  weight,  we  may  regard 
that  weight  as  resting  upon  one  of  the  helices,  and  sustained 
in  equilibrium  by  a  force  applied  horizontally.  If  we  suppose 
the  supporting  helix  to  be  developed  on  a  vertical  plane,  it 
will  form  an  inclined  plane,  whose  base  is  the  circumference 
of  the  base  of  the  cylinder  on  which  it  lies,  and  whose  alti- 
tude is  the  distance  between  the  threads  of  the  screw. 

Let  AB  represent  the  development  of  this  helix  on  a 
vertical  plane,  and  denote  by  F  the  force  applied  parallel  to 
the  base,  and  immeeliately  to  the  weight  11,  to  sustain  it 
on  the  plane.     We  shall  have  (Art  85), 

F:  E  ::  BC  :  AC. 


ELEMENTARY    MACHINES. 


125 


But  the  power  is  actually 
applied  through  the  medium 
of  a  lever.  Denoting  the 
radius  OG  of  the  cylinder  of 
the  supporting  helix,  by  r,  and 
the  arm  of  lever  of  the  power 
P  by  p,  we  shall  have,  from 
the  principle  of  the  lever, 

Pi  F::  r  :  »: 


i 


Fig.  87. 


or, 


P  :  F :  :  2*r  :  2«p. 


Combining  this  proportion  with  the  preceding  one,  and 
recollecting  that  AG  =  2<xr,  we  deduce  the  proportion, 


P:  B  ::  BG  :  2*p . 


(48.) 


That  is,  the  power  is  to  the  resistance  as  the  distance  be- 
tween the  threads  is  to  the  circumference  described  by  the 
point  of  application  of  the  power. 

By  suitably  diminishing  the  distance  between  the  threads, 
other  things  being  equal,  any  amount  of  mechanical  ad- 
vantage  may  be  obtained. 

The  screw  is  used  for  producing  great  pressures 
through  very  small  distances,  as  in  pressing  books  for  the 
binder,  packing  merchandise,  expressing  oils,  and  the  like. 
On  account  of  the  great  amount  of  friction,  and  other  hurt- 
ful resistances  developed,  the  modulus  of  the  machine  is 

very  small. 

The  Differential  Screw. 

98.  The  differential  screw  consists  essentially  of  an  ordi- 
nary screw,  as  just  described,  into  the  end  of  which  works 
a  smaller  screw,  having  its  axis  coincident  with  the  first, 
but  having  its  thread  turned  in  a  contrary  direction ;  that 
is,  it  is  what  is  technically  called  a  left-handed  screw,  the 
first  screw  being  a  right-handed  one.  The  distance  between 
the  threads  of  the  second  screw  is  somewhat  less  than  that 
between  the  threads  of  the  first  screw,  and  this  difference 


126 


MECHANICS. 


may  be  made  as  small  as  desirable.  The  second  screw  is  so 
arranged  that  it  admits  of  a  longitudinal  motion,  but  not  of 
a  motion  of  rotation.  By  the  action  of  the  differential  screw, 
the  weight  is  raised  vertically  through  a  distance  equal  to 
the  difference  of  the  distances  between  the  threads  on  the 
two  screws,  for  each  revolution  of  the  point  of  application 
of  the  power.  For,  were  the  first  screw  alone  to  turn,  the 
weight  would  be  raised  through  a  distance  equal  to  the  dis- 
tance between  its  threads  ;  but,  because  the  second  screw  is 
a  left-handed  one,  this  distance  will  be  diminished  by  a  dis- 
tance equal  to  that  between  its  threads.  We  may,  there- 
fore, write  the  following  rule : 

The  power  is  to  the  resistance  as  the  difference  of  the 
distances  between  the  threads  of  the  two  screws  is  to  the  cir- 
cumference described  by  the  point  of  application  of  the 
power. 

Endless  Screw. 

99.  The  endless  screw  is  a  screw  secured  by  shoulders, 
so  that  it  cannot  be  moved  longi- 
tudinally, and  working  into  a 
toothed  wheel.  The  distance  be- 
tween the  teeth  should  be  nearly 
equal  to  the  distance  between 
the  threads  of  the  screw.  When 
the  screw  is  turned,  it  imparts  a 
rotary  motion  to  the  wheel,  which 
may  be  utilized  by  any  mechani- 
cal device.  The  conditions  of 
equilibrium  are  the  same  as  for 
the  screw,  the  resistance  in  this 
case  being  offered  by  the  wheel, 
in  the  direction  of  its  circumference. 

Machines  of  this  kind  are  used  in  determining  the  number 
of  revolutions  of  an  axis.  An  endless  screw  is  arranged  to 
turn  as  many  times  as  the  axis,  and  being  connected  with  a 
train  of  light  wheel-work,  the  last  piece  of  which  bears  an 
index,  the  number  of  revolutions  can  readily  be  ascertained 


Fig.  88. 


ELEMENTARY    MACHINES.  127 

at  any  instant.  As  an  example,  suppose  the  first  wheel  to 
have  100  teeth,  and  to  bear  on  its  arbor  a  smaller  wheel, 
having  10  teeth  ;  suppose  this  wheel  to  engage  with  a  larger 
wheel  having  100  teeth,  and  so  on.  When  the  endless 
screw  has  made  10,000  revolutions,  the  first  wheel  will  have 
made  100  revolutions,  the  second  large  wheel  will  have 
made  10  revolutions,  and  the  third  wheel  1  revolution.  By 
a  suitable  arrangement  of  indices,  the  exact  number  of  revo- 
lutions of  the  axis,  at  any  instant,  may  be  read  off  from  the 
instrument. 

EXAMPLES. 

1.  What  must  be  the  distance  between  the  threads  of  a 
screw  in  order  that  a  power  of  28  lbs.,  acting  at  the  ex- 
tremity of  a  lever  25  inches  long,  may  sustain  a  weight  of 
10,000  lbs.  ?  Arts.  .4396  inches. 

2.  The  distance  between  the  threads  of  a  screw  is  i  of  an 
inch.  What  resistance  can  be  supported  by  a  power  of 
60  lbs.,  acting  at  the  extremity  of  a  lever  15  inches  long  ? 

Arts.   16,964  lbs. 

3.  The  distance  from  the  axis  of  the  trunions  of  a  gun 
weighing  2,016  lbs.  to  the  elevating  screw  is  3  feet,  and  the 
distance  of  the  centre  of  gravity  of  the  gun  from  the  same 
axis  is  four  inches.  If  the  distance  between  the  threads  of 
the  screw  be  f  of  an  inch,  and  the  length  of  the  lever  5  inches, 
what  power  must  be  applied  to  sustain  the  gun  in  a  horizon- 
tal position?  Ans.  4.754  lbs. 

The  Wedge. 

100.      The  wedge  is  a  solid,  bounded  by  a   rectangle 
1?D,  called  the  back  ;  two  equal  rect- 
angles, AF  and  DF,  called  faces ;  < 
and  two  equal  isosceles  triangles,  called 
ends.     The   line   EF,    in    which   the                D, 
faces  meet,  is  called  the  edge. 


M 


The  power  is  applied  at  the  back, 
to  which  its  direction  should  be 
normal,  and  the  resistance  is  applied 
to  the  faces,  and  in  directions  normal  E  ' 

to  them.     One  half  of  the  resistance  Fis' 89' 


128  MECHANICS. 

is  applied  normally  to  one  face,  and  the  other  half  normally 
to  the  other  face.  Let  ABC  be  a 
section  of  a  wedge  made  by  a  plane 
at  right  angles  to  the  edge.  Denote 
the  power  by  P,  and  the  resistance 
opposed  to  each  face  by  ±R;  denote 
the  angle  BAC  of  the  wedge  by 
'lz.  Produce  the  directions  of  the 
resistances  till  they  intersect  in  0. 
This  point  will  be  on  the  line  of  direc- 
tion of  the  power.  Lay  off  OF  to 
represent    the   power,  and    complete 

the  parallelogram  EB;  then  will  OE  and  OB  repre- 
sent the  resistances  developed  by  the  power.  Let  each 
of  the  forces  \B  be  resolved  into  two  components,  one  per- 
pendicular to  OE,  and  the  other  coinciding  with  it.  The 
two  former  will  be  equal  and  directly  opposed  to  each  other, 
whilst  the  two  latter  will  hold  the  force  P  in  equilibrium. 
Since  BE  is  perpendicular  to  FO,  and  BO  perpen- 
dicular to  GA,  the  angle  OBE  is  equal  to  the  angle 
OA  C,  or  z.  The  component  of  \R  in  the  direction  of 
OE,    is  i/iMn;  ;    hence,   twice  this,   or  i?sin?  =  P.     But 

sins  =  y-       =  y-,  in  which  b  denotes  the  breadth  of  the 

back  JBO,  and  I  the  length  of  the  face  CA.  Substituting 
this  expression  for  sinp,  and  reducing,  we  have, 

B  X  ib  =  PI,    or   P  :  B  :  :  ±h  :  I     .     ( 49.) 

That  is,  the  power  is  to  the  resistance  as  one-half  of  the 
breadth  of  the  back  is  to  the  length  of  the  face  of  the  wedge. 

The  mechanical  advantage  of  the  wedge  maybe  increased 
by  diminishing  the  breadth  of  the  bark,  or,  in  other  woi-ds, 
by  making  the  edge  sharper.  The  principle  of  the  wedge 
finds  an  Important  application  in  all  cutting  instruments,  as 
knives,  razors,  and  the  like.  By  diminishing  the  thickness 
of  the  back,  the  instrument  is  rendered  liable  to  break, 
hence  the  necessity  of  forming  cutting  instruments  of  the 
hardest  and  most  tenacious  materials. 


ELEMENTARY    MACHINES.  129 

General  remarks  on  Elementary  Machines. 
101.  We  have  thus  far  supposed  the  power  and  resist- 
ance to  be  in  equilibrium,  through  the  intervention  of  the 
machine,  their  points  of  application  being  at  rest.  If  we 
now  suppose  the  point  of  application  to  be  moved  through 
any  distance,  by  the  action  of  an  extraneous  force,  the  point 
ol*  application  of  the  power  will  move  through  a  correspond- 
ing space.  These  spaces  will  be  described  in  conformity 
with  the  design  of  the  machine  ;  and  it  will  be  found,  in 
each  instance,  that  they  are  inversely  proportional  to  the 
forces.  If  we  suppose  these  spaces  to  be  infinitely  small, 
they  may,  in  all  cases,  be  regarded  as  straight  lines,  which 
will  also  be  the  virtual  velocities  of  the  forces.  If  the  point 
of  application  moves  in  a  direction  contrary  to  the  direction 
of  the  resistance,  the  point  of  application  of  the  power  will 
move  in  the  direction  of  the  power.  If  we  denote  the  paths 
described  by  those  points  respectively,  by  or,  and  Sp,  we 
shall  have, 

Pop  —  JlSr  =  0;    or  PSp  —  Eor  .   .     (50.) 

That  is,  the  algebraic  sum  of  the  virtual  moments  is  equal 
to  0.  Or,  we  might  enunciate  the  principle  in  another  man- 
ner, by  saying,  that  in  all  cases,  the  quantity  of  work  of  the 
power  is  equal  to  the  quantity  of  work  of  the  resistance. 

We  shall  illustrate  this  principle,  by  considering  a  single 
case,  that  of  the  single  movable  pulley,  leaving  its  further 
application  to  the  remaining  machines,  as  exercises  for  the 
student. 

In  the  figure,  suppose  that  an  extraneous 
force  acts  to  raise  the  resistance  P,  through 
the  infinitely  small  space  DE,  denoted  by  Sr ; 
the  point  of  application  of  P  must  be  raised 
through  the  infinitely  small  space  FG,  denoted 
by  8p,  in  order  that  the  equilibrium  may  be 
preserved. 

In  order  that  the  resistance  may  be  .raised 
through  the  distance  DE,  both  branches  of  the 
rope  enveloping  the  pulley  must  be  shortened         F,*M- 
by  the  same  amount ;  or,   what  is  the  same 
6* 


130  MECHANICS. 

thing,  tin;  free  end  of  the  rope  must  ascend  through  twice 
the  distance  DE.     Hence, 

§p  =  2  or. 
But,  from  the  conditions  of  equilibrium, 

P  =  JJB. 
Multiplying  these  equations,  member  by  member,  we  have, 
PSp  =  Mr. 

Hence,  the  principle  is  proved  for  this  particular  case.  In 
like  manner,  it  may  be  shown  to  hold  good  for  all  of  the 
elementary  machines. 

The  principle  of  equality  of  work  of  the  power  and  resist- 
ance being  true  for  any  infinitely  short  time,  it  must  neces- 
sarily hold  good  for  any  time  whatever.  Hence,  we  con- 
clude, that  the  quantity  of  work  of  the  power,  in  overcoming 
any  resistance,  is  equal  to  quantity  of  work  of  the  resist- 
ance. Although,  by  the  application  of  a  very  small  power, 
we  are  able  to  overcome  a  very  great  resistance,  the  space 
passed  over  by  the  point  of  application  of  the  power  must 
be  as  much  greater  than  that  passed  over  by  the  point  of 
application  of  the  resistance,  as  the  resistance  is  greater 
than  the  power.  This  is  generally  expressed  by  saying, 
that  what  is  gained  in  ])ower  is  lost  in  velocity. 

We  see,  therefore,  that  no  power  is,  or  can  be,  gained ; 
the  only  function  of  a  machine  being  to  enable  a  smaller 
force  to  accomplish  in  a  longer  time,  what  a  larger  force 
would  be  required  to  perform  in  a  shorter  time. 

Friction. 
102.  Friction  is  the  resistance  which  one  body  experi- 
ences in  moving  upon  another,  the  two  being  pressed 
together  by  some  force.  This  resistance  arises  from 
inequalities  in  the.  two  surfaces,  the  projections  of  one  sur- 
face sinking  into  the  depressions  of  the  other.  In  order  to 
overcome  this  resistance,  a  sufficient  force  must  be  applied 


HURTFUL    RESISTANCES.  1  3  1 

to  break  off,  or  bend  down,  the  projecting  points,  or  else  to 
lift  the  moving  body  clear  of  the  inequalities.  The  force 
thus  applied,  is  equal,  and  directly  opposed  to  the  force  of 
friction,  which  is  tangential  to  the  two  surfaces.  The  force 
which  presses  the  surfaces  together,  is  normal  to  them  both 
at  the  point  of  contact. 

Friction  is  distinguished  as  sliding  and  rolling.  The  for- 
mer arises  when  one  body  is  drawn  upon  another ;  the  lat- 
ter when  one  body  is  rolled  upon  another.  In  the  case  of 
rolling  friction,  the  motion  is  such  as  to  lift  the  projecting 
points  out  of  the  depressions  ;  the  resistance  is,  therefore, 
much  less  than  in  sliding  friction. 

Between  certain  bodies,  the  friction  is  somewhat  different 
when  motion  is  just  beginning,  from  what  it  is  when  motion 
has  been  established.  The  friction  developed  when  a  body 
is  passing  from  a  state  of  rest  to  a  state  of  motion,  is  called 
friction  of  quiescence  ;  that  which  exists  between  bodies  in 
motion,  is  called  friction  of  motion. 

The  following  laws  of  friction  have  been  established  by 
numerous  experiments,  viz. : 

First,  the  friction  of  quiescence  between  the  same  bodies, 
is  proportional  to  the  normal  pressure,  and  independent  of 
the  extent  of  the  surfaces  in  contact. 

Secondly,  the  friction  of  motion  between  the  same  bodies, 
is  proportional  to  the  normal  pressure,  and  independent, 
both  of  the  extent  of  the  surfaces  in  contact,  and  of  the 
velocity  of  the  moving  body. 

Thirdly,  for  compressible  bodies,  the  friction  of  quiescence 
is  greater  than  the  friction  of  motion  ;  for  bodies  which 
are  sensibly  incompressible,  the  difference  is  scarcely  appre- 
ciable. 

Fourthly,  friction  may  be  greatly  diminished,  by  intcr- 
posing  unguents  between  the  rubbing  surfaces. 

Unguents  serve  to  fill  up  the  cavities  of  surfaces,  and  thug 
to  diminish  the  resistances  arising  from  their  roughness. 
For  slow  motions  and  great  pressures,  the  more  consistent 
unguents  are  used,  as  lard,  tallow,  and  various  mixtures  • 


E 


132  MECHANICS. 

for  rapid  motions,  and  light  pressures,  oils  are  generally  ein« 
ployed. 

The  ratio  obtained  by  dividing  the  entire  force  of  friction 
by  the  normal  pressure,  is  called  the  coefficient  of  friction  ; 
the  value  of  the  coefficient  of  friction  for  any  two  substances, 
»nay  be  determined  experimentally  as  follows  : 

Let  AB  be  a  horizontal  plane 
formed  of  one  of  the  substances, 
and  let  0  be  a  cubical  block  of 
the  other  substance  resting 
upon  it.  Attach  a  string  OC\ 
jV  to  the  block,  so  that  its  direc- 
Fig.  92.  tion  shall  pass  through  its  cen- 

tre of  gravity,  and  be  parallel 
to  AB  ;  let  the  string  pass  over  a  fixed  pulley  (7,  and  let  a 
weight  F,  be  attached  to  its  extremity. 

Increase  the  weight  F  till  the  body  0  just  begins  to 
slide  along  the  plane,  then  will  this  weight  measure  the 
whole  force  of  friction.  Denote  this  weight  by  F,  that  of 
the  body,  or  the  normal  pressure,  by  P,  and  the  coefficient 
of  friction,  by  f     Then,  from  the  definition,  we  shall  have, 

1         P 

In  this  manner,  values  for  f  corresponding  to  different 
substances,  may  be  found,  and  arranged  in  tables.  This 
experiment  gives  the  friction  of  quiescence.  If  the  weight 
F  is  such  as  to  keep  the  body  0  in  uniform  motion,  the 
resulting  value  of/'  will  correspond  to  friction  of  motion. 

The  value  of /J  for  any  substance,  is  called  the  unit,  or 
coefficient  of  friction.  Hence,  we  may  define  the  unit,  or 
coefficient  of  friction,  to  be  the  friction  <!>/<  to  a  normal 
pn  8i  ure  of  ont  pound. 

Having  given  the  normal  pressure  in  pounds,  and  the 
unit  of  friction,  the  entire  friction  will  be  found  bv  multi 
plying  these  quantities  together. 


HURTFUL    RESISTANCES.  133 

There  is  a  second  method  of  finding  the  value  of/  ex- 
perimentally, as  follows : 

Let  AB  be  an  inclined  plane,  formed  of  one  of  the  sub- 
stances, and  0  a  cubical  block, 
formed  of  the  other  substance, 
and  resting  upon  it.  Elevate  the 
plane  till  the  block  just  begins  to 
slide  down  the  plane  by  its  own 
weight.  Denote  the  angle  of  in- 
clination, at  this  instant,  by  a,  and  Fig.  93. 
the  weight  of  0,  by  W.    Resolve 

the  force  W  into  two  components,  one  normal  to  the  sur 
face  of  the  plane,  and  the  other  one  parallel  to  it.  Denote 
the  former  component  by  P,  and  the  latter  by  Q.  Since 
0  W  is  perpendicular  to  A  C,  and  OP  to  AB,  the  angle 
WOP  is  equal  to  a.     Hence, 

P  =  TFcosa,     and    Q  =  TFsina. 

The  normal  pressure  being  equal  to  TFcosa,  and  the  force 
of  friction  being  TFsina,  we  shall  have,  from  the  principles 
already  explained, 

TFsina  BO 

J  TFcosa  A  0 

The  angle  a  is  called  the  angle  of  friction. 

Limiting  Angle  of  Resistance. 
103.     Let  AB  be  any  plane  surface,  and  0  a  body  rest- 
ing upon  it.    Let  B  1  >e  the  resultant 
of  all  the  forces  acting  upon  it,  in-  Q 

eluding  the  weight  applied  at  the  ^ -.,.     "/*l 

centre  of  gravity.  Denote  the  angle  /       \ 

between  R  and  the  normal  to  AB,  H  *p 

by  a,  and  suppose  R  to  be  resolved  Fi    94> 

into  two  components  P  and  §,  the 

former  parallel  to  AB,  and  the  latter  perpendicular  to  it ; 

we  alial1  have, 

P  =  i^sina,     and    Q  =  Rcosc. 


134 


MECHANICS. 


The  friction  due  to  the  normal  pressure  will  be  equal  to 
f-Rcosji.  Now,  when  the  tangential  component  jR*inx  is 
less  than  /Vicos*,  the  body  will  remain  at  rest ;  when  it  is 
greater  than  /ifoos*,  the  body  will  slide  along  the  plane ; 
and  when  the  two  are  equal,  the  body  will  be  in  a  state 
bordering  on  motion  along  the  plane.  Placing  the  two 
equal,  we  have, 

fltcosoi  =  Ushvx  ;         .'.    f  =  tan/. 

The  value  of  a  is  called  the  limiting  angle  of  resistance, 
and    is  equal  to  the  inclination    of  the 

plane,  when  .the  body  is  about  to  slide  ^ __^_ 

down  by  its  own  weight.     If,  now,  the  ^^_L~-/R 

line  Oil  be  revolved  about  the  normal,  it  \    '  w 

will   generate   a   conical   surface,  within  /    \/      7 

which,  if  any  force  whatever,  including         /         o  / 
the  weight,  be  applied  at  the  centre  of  Fig.  95. 

gravity,  the  body  will  remain  at  rest,  and 
without  which,  if  a  sufficient  force  be  applied,  the  body  will 
slide  along  the  plane.     This  cone  is  called  the  limiting  cone 
of  resistance. 

The  values  of/",  or  the  coefficient  of  friction,  in  some  of  the 
most  common  cases,  as  determined  by  Morix,  is  appended  : 

TABLE. 

Bodies  between  which  friction  takes  place.  Coefficient  of  friction. 

Iron  on  oak, .62 

Cast  iron  on  oak, .49 

Oak  on  oak,  fibres  parallel,  ....  .48 

Do.,  do.,  greased, .10 

Cast  iron  on  cast  iron, .15 

Wrought  iron  on  wrought  iron,  .     .  .14 

Brass  on  iron, .16 

Brass  on  brass, .20 

Wrought  iron  on  cast  iron,  .     .         .  .19 

Cast  iron  on  elm, .19 

Soft  limestone  on  the  same,  ....  .64 

Hard  limestone  on  the  same,          .     .  .38 


HURTFUL    RESIST ANCF.S.  135 

Bodies  between  ichich  friction  takes  place.  Coefficient  of  friction. 

Leather  belts  on  wooden  pulleys,        .  .47 

Leather  belts  on  cast  iron  pulleys,       .  .28 

Cast  iron  on  cast  iron,  greased,       .     .  .10 

Pivots  or  axes  of  wrought  or  cast  iron,  on  brass  or  cast- 
iron  pillows : 

1st,  when  constantly  supplied  with  oil,  .05 

2nd,  when  greased  from  time  to  time,  .08 

3rd,  without  any  application,     ...  .15 

Rolling  Friction. 

104.  Rolling  friction  is  the  resistance  which  one  body 
offers  to  another  when  rolling  along  its  surface,  the  two 
being  pressed  together  by  some  force.  This  resistance,  like 
that  in  sliding  friction,  arises  from  the  inequalities  of  the 
two  surfaces.  The  coefficient,  or  unit,  of  rolling  friction  is 
equal  to  the  quotient  obtained  by  dividing  the  entire  force 
of  friction  by  the  normal  pressure.  This  coefficient  is  much 
less  than  the  coefficient  of  sliding  friction. 

The  following  laws  of  friction  have  been  established, 
when  a  cylindrical  body  or  wheel  rolls  upon  a  plane  : 

First,  the  coefficient  of  rolling  friction  is  proportional  to 
the  normal  pressure : 

Secondly,  it  is  inversely  proportioned  to  the  diameter  of 
the  cylinder  or  to  heel: 

Thirdly,  it  increases  as  the  surface  of  contact  and  velocity 
increase. 

In  many  cases  there  is  a  combination  of  both  sliding  and 
rolling  friction  in  the  same  machine.  Thus,  in  a  car  upon  a 
railroad-track,  the  friction  at  the  axle  is  sliding,  and  that 
between  the  circumference  of  the  wheel  and  the  track  is 
rolling. 

Adhesion. 

105.  Adhesion  is  the  resistance  which  one  body  ex- 
periences in  moving  upon  another  in  consequence  of  the 
cohesion  existing;  between  the  molecules  of  the  surf  ices  in 
contact.     This  resistance  increases  when  the  surfaces  are 


136 


MECHANICS. 


allowed  to  remain  for  some  time  in  contact,  and  is  very 
slight  when  motion  lias  been  established.  Both  theory  and 
experiment  show  that  adhesion  between  the  same  surfaces  is 
proportional  to  the  extent  of  the  surface  of  contact. 

The  coefficient  of  adhesion  is  the  quotient  obtained  by 
dividing  the  entire  adhesion  by  the  area  of  the  surface  of 
contact.  Or,  denoting  the  entire  adhesion  by  A,  the  area 
of  the  surface  of  contact  by  Sy  and  the  coefficient  of  adhesion 
by  a,  we  have, 

A 


a=  V 


A  =  aS. 


To  find  the   entire    adhesion,    we  multiply   the   unit   of 
adhesion  by  the  area  of  the  surface  of  contact. 


Stiffness  of  Cords. 

106.  Let  0  represent  a  pulley,  with  a  cord  AJB^ 
wrapped  around  its  circumference,  and 
suppose  a  force  P,  applied  at  i?,  to  over- 
come the  resistance  i?,  and  impart  motion 
to  the  pulley.  As  the  rope  winds  upon  the 
pulley,  at  (7,  its  rigidity  acts  to  increase  the 
arm  of  lever  of  i?,  and  to  overcome  this 
resistance  to  flexure  an  additional  force  is 
required.  For  the  same  pulley,  this  addi- 
tional force  may  be  represented  by  the 
algebraic  expression, 

a  -f-  bll, 


Fig.   96. 


in  which  a  and  b  are  constants  dependent  upon  the  nature 
and  construction  of  the  rope,  and  li  is  the  resistance  to 
be  overcome,  or  the  tension  of  the  covd.  AC.  The  values  of 
a  and  b  for  different  ropes  have  been  ascertained  by  experi- 
ment, and  tabulated.  Finally,  if  the  *-\\uv  rope  be  vround 
upon  pulleys  of  different  diameters,  the  additional  force  is 
found  to  vary  inversely  as  their  diameters.  If  the  diameter 
of  .lie  pulley  be  denoted  by  D,  and  the  resistance  due  to 
stiffness  of  cordage  be  denoted  by  S,  we  shall  have, 


HURTFUL    RESISTANT.  ES.  137 

Q       a  +  bE 
S  =  — g~. 

In  the  case  of  the  pulley,  if  we  neglect  friction,  we  shall 
have,  when  the  motion  is  uniform, 

a  +  bJR 
P  =  J2  +  — p-, 

for  the  algebraic  expression  of  the  conditions  of  equilibrium. 
The  values  of  a   and  b  have  been   determined   experi- 
mentally for  all  values  of  R  and  Z>,  and  tabulated. 

Atmospheric  Resistance. 

107.  The  atmosphere  exercises  a  powerful  resistance  to 
the  motion  of  bodies  passing  through  it.  This  resistance  is 
due  to  the  inertia  of  the  particles  of  air,  which  must  be 
overcome  by  the  force  of  a  moving  body.  It  is  evident,  in 
the  first  place,  other  things  being  equal*  that  the  resistance 
will  depend  upon  the  amount  of  surface  of  the  moving  body 
which  is  exposed  to  the  air  in  the  direction  of  the  motion. 
In  the  second  place,  the  resistance  must  increase  with  the 
square  of  the  velocity  of  the  moving  body ;  for,  if  we  sup- 
pose the  velocity  to  be  doubled,  there  will  be  twice  as  many 
particles  met  with  in  a  second,  and  each  particle  will  collide 
against  the  moving  body  with  twice  the  force,  hence ;  if  the 
velocity  be  doubled,  the  resistance  will  be  quadrupled.  By 
a  similar  course  of  reasoning,  it  may  be  shown  that,  if  the 
velocity  be  tripled,  the  retardation  will  become  nine  times  as 
great,  and  so  on.  If,  therefore,  the  retardation  correspond- 
ing to  a  square  foot  of  surface,  at  any  given  velocity,  be 
determined,  the  retardation  corresponding  to  any  surface 
and  any  velocity  whatever  may  be  computed. 

Influence  of  Friction  on  the  Inclined  Plane. 

108.  Let  it  be  required  to  determine  the  relation 
between  the  power  and  resistance,  when  the  power  is  just 
on  the  point  of  imparting  motion  to  a  body  up  an  inclined 
plane,  friction  being  taken  into  account. 


138  MECHANICS. 

Let  AB  represent  the  plane,  0  the  body,  OP  the  power 
on  the  point  of  imparting  motion 
up  the  plane,  and  OR  the  weight 
of  the  body.  Denote  the  power 
by  P,  the  weight  by  R,  the  in- 
clination of  the  plane  by  a,  and 
the  angle  between  the  direction 
of  the  power  and  the  normal  to 
the  plane  by  (3.  Let  P  and  R 
be  resolved  into  components  re-  Fi    97 

spectively  parallel  and  perpendi- 

dicular  to  the  plane.  We  shall  have,  for  the  parallel  com- 
ponents, Rsina  and  Psin/3,  and  for  the  perpendicular  com- 
ponents, Rcosol  and  Pcos/3.  The  resultant  of  the  normal 
components  will  be  equal  to  Rcosol  —  Pcos3 ;  and,  if  we 
denote  the  coefficient  of  friction  by  /,  we  shall  have  for  the 
entire  force  of  friction  (Art.  102), 

/(Rcosx  —  Pcos/3). 

When  we  consider  the  body  on  the  eve  of  motion  up  the 
plane,  the  component  jPsin/3  must  be  equal  and  directly 
opposed  to  the  resultant  of  the  force  of  friction  and  the 
component  Psina  ;  hence,  we  must  have, 

Psin-S  —  Psina  +■  f  (Reosx  —  PcosQ). 

Performing  the  multiplications  indicated,  and  reducing, 
we  have, 

j>  =  Ji\»™+Jp±\  .  .  .  (51.) 

(  sin>5  -f-/cos,3  j  ' 

If  we  suppose  an  equilibrium  to  exist,  the  body  being  on 
on  the  eve  of  motion  down  the  plane,  we  shall  have. 

Psin3  +/(7vcosa  —  Pcos/3)  =  Psina. 

Whence,  bv  reduction, 

T  =  RVT- f™\\    .     .     .     (62.1 

/  sin/3  —  /cos/3  |  v 


HURTFUL    RESISTANCES.  139 

From  these  expressions,  two  values  of  P  may  be 
found,  when  a,  /3,  f\  and  R  are  given.  It  is  evident  that 
any  value  of  P  greater  than  the  tirst  will  >«*use  the  body  to 
slide  up  the  plane,  that  any  value  less  than  the  second  yill 
permit  it  to  slide  down  the  plane,  and  that  for  any  inter- 
mediate value  the  body  will  remain  at  rest  on  the  plane. 

If  we  suppose  P  to  be  parallel  to  the  plane,  we  shall  have 
sin/3  r=  1,   cos/3  =  0,   and  the  two  values  of  P  reduce  to 

P  =  jR(sina  +  /cosa)     .     .     .      ( 53.) 
and, 

P  =  R(sma  -  /cosa)      .     .      .      ( 54.) 

If  friction  be  neglected,  we  have  /  =  0;  whence,  by 
substitution, 

p        *>•  P        BO 

a    result  which  agrees  with  that  deduced  in  a  preceding 
article. 

To  find  the  quantity  of  work  of  the  power  whilst  drawing 
a  body  up  the  entire  length  of  the  inclined  plane,  it  may 
be  observed  that  the  value  of  P,  in  Equation  (53),  is  equal 
to  that  required  to  maintain  the  body  in  uniform  motion 
after  motion  has  commenced. 

Multiplying  both  members  of  that  equation  by  AJB,  we 
have, 

P  x  AB  =  R  x  AB  sin  a  +  fR  x  AB  cos* 

=  R  x  BC  +fR  x  AC. 

But  R  x  B  C  is  the  quantity  of  work  necessary  to  raise 
the  body  through  the  vertical  height  BC;  and  fR  x  AC, 
is  the  quantity  of  work  necessary  to  draw  the  body  horizon- 
tally through  the  distance  A  C  (Art.  75).  Hence,  the  quan- 
tity of  work  required  to  draw  a  body  up  an  inclined  plane, 
when  the  power  is  parallel  to  the  plane,  is  equal  to  the  quan- 
tity of  work  necessary  to  draw  it  horizontally  across  the 
base  of  the  plane,  plus  the  quantity  of  work  necessary  to 
raise  it  vertically  through  the  height  of  the  plane. 


140  MECHANICS. 

A  curve  situated  in  a  vertical  plane  may  be  regarded  as 
made  up  of  an  infinite  number  of  inclined  planes.  We 
infer,  therefore,  that  the  quantity  of  work  necessary  to  draw 
a  body  up  a  curve,  the  power  acting  always  parallel  to  the 
direction  of  the  curve,  is  equal  to  the  quantity  of  work  ne- 
cessary to  draw  the  body  over  the  horizontal  projection  of 
the  curve,  plus  the  quantity  of  work  necessary  to  raise  th-3 
body  through  a  height  equal  to  the  difference  of  altitude  of 
the  two  extremities  of  the  curve. 

The  last  two  principles  enable  us  to  compare  the  quanti- 
ties of  work  necessary  to  draw  a  train  of  cars  over  a  hori- 
zontal track,  and  up  an  inclined  track,  or  a  succession  of 
inclined  tracks.  We  may,  therefore,  compute  the  length  of 
a  horizontal  track  which  will  consume  the  same  amount  of 
work,  furnished  by  the  motor,  as  is  actually  consumed  in 
consequence  of  the  undulation  of  the  track. 

We  are  thus  enabled  to  compare  the  relative  advantages 
of  different  proposed  routes  of  railroad,  with  respect  to  the 
motive  power  required  for  working  them. 

Line  of  Least  Traction. 

109.  The  force  employed  to  draw  a  body  with  uniform 
motion  along  an  inclined  plane,  is  called  the  force  of  trac- 
tion /  and  the  line  of  direction  of  this  force  is  the  line  of 
traction.  In  Equation  (51),  P  represents  the  force  of  trac- 
tion required  to  keep  a  body  in  uniform  motion  up  an 
inclined  plane,  and  (3  is  the  angle  which  the  line  of  traction 
makes  with  the  normal.  It  is  plain,  that  when  f3  varies,  other 
things  being  the  same,  the  value  of  P  will  vary ;  there  will 
evidently  be  some  value  of  3,  which  will  render  P  the  least 
possible  ;  the  direction  of  P  in  this  case,  is  called  the  line  of 
least  traction;  and  it  is  along  this  line  that  a  force  can  be 
applied  with  greatest  advantage,  to  draw  a  body  up  an 
inclined  plane.  If  we  examine  the  expression  for  P,  in 
Equation  (51),  we  see  that  the  numerator  remains  constant ; 
therefore,  the  expression  for  P  will  be  least  possible  when 
the  denominator  is  the  greatest  possible.     By  a  simple  pro- 


HURTFUL    RESISTANCES.  141 

cess  of  the  Differential  Calculus,  it  may  be  shown  that  the 

denominator  will  be  the  greatest  possible,  or  a  maximum, 

when, 

/  =  cot  o,     or  /  -  tan  (90°  —  /8). 

That  is,  the  power  will  be  applied  most  advantageously, 
when  it  makes  an  angle  with  the  inclined  plane  equal  to  the 
angle  of  friction. 

From  the  second  value  of  jP,  it  may  be  shown,  in  like 
maimer,  that  a  force  will  be  most  advantageously  applied,  to 
prevent  a  body  from  sliding  down  the  plane,  when  its  direc- 
tion makes  an  angle  with  the  plane  equal  to  the  supplement 
of  the  angle  of  friction,  the  angle  being  estimated  as  before 
from  that  part  of  the  plane  lying  above  the  body. 

Friction  on  an  Axle. 

110.     Let  it  be  required  to  determine  the  position  of 
equilibrium  of  a  horizontal  axle,  resting  in 
a  cylindrical  box,  when  the  power  is  just 
on  the  point    of  overcoming  the  friction 
betwi  en  the  axle  and  box. 

Let  0'  be  the  centre  of  a  cross  section 
of  the  axle,  0  the  centre  of  the  cross  sec- 
tion of  the  box,  and  N  their  point  of  con- 
tact, when  the  power  is  on  the  point  of 
overcoming  the  friction  between  the  axle 
and  box.  The  element  through  iVwill  be  the  line  of  con- 
tact of  the  axle  and  box. 

When  the  axle  is  only  acted  upon  by  its  own  weight,  the 
element  of  contact  will  be  the  lowest  element  of  the  box. 
If,  now,  a  power  be  applied  to  turn  the  axle  in  the  direction 
indicated  by  the  arrow-head,  the  axle  will  roll  up  the  inside 
of  the  box  until  the  resultant  of  all  the  forces  acting  upon 
it  becomes  normal  to  the  surface  of  the  axle  at  some  point 
of  the  element  through  X.  This  normal  force  pressing  the 
axle  against  the  box,  will  give  rise  to  a  force  of  friction  act- 
ing tangentially  upon  the  axle,  which  will  be  exactly  equal 
to  the  tangential  force  applied  at  the  circumference  of  the 


142  MECHANICS. 

axle  to  produce  rotation.  If  the  axle  be  rolled  further  up 
the  side  of  the  box,  it  will  slide  back  to  N\  if  it  be  moved 
down  the  box,  it  will  roll  back  to  iV,  under  the  action  of  the 
force.  In  tins  position  of  the  axle,  it  is  in  the  condition  of 
a  body  resting  upon  an  inclined  plane,  just  on  the  point  of 
sliding  down  the  plane,  but  restrained  by  the  force  of  fric- 
tion. Hence,  if  a  plane  be  passed  tangent  to  the  surface  of 
the  box,  along  the  element  jV,  it  will  make  with  the 
horizon  an  angle  equal  to  the  angle  of  friction.  The  rela- 
tion between  the  power  and  resistance  may  then  be  fc  und, 
98  in  Art.  108. 


RECTILINEAR   MOTION.  143 


CHAPTER  V. 

RECTILINEAR   AND    PERIODIC    MOTION. 
Motion. 

111.  A  material  point  is  in  motion  when  it  continually 
changes  its  position  in  space.  When  the  path  of  the  moving 
point  is  a  straight  line,  the  motion  is  rectilinear ;  when  it  is 
a  curved  line,  the  motion  is  curvilinear.  When  the  motion 
is  curvilinear,  we  may  regard  the  path  as  made  up  of  infi- 
nitely short  straight  lines  ;  that  is,  we  may  consider  it  as  a 
polygon,  whose  sides  are  infinitely  small.  If  any  side  of  this 
polygon  be  prolonged  in  the  direction  of  the  motion,  it  will 
be  a  tangent  to  the  curve.  Hence,  we  say,  that  a  point 
always  moves  in  the  direction  of  a  tangent  to  its  path. 

Uniform  Motion. 

112.  Uniform  motion  is  that  in  which  the  moving 
point  describes  equal  spaces  in  any  arbitrary  equal  portions 
of  time.  If  we  denote  the  space  described  hi  one  second 
by  v,  and  the  space  described  in  t  seconds  by  s,  we  shall 
have,  from  the  definition, 

s  =  vt ;         .'.     v  =  -     .     .     .     (55.) 

0 

From  the  first  of  these  equations,  we  see  that  the  space 
described  in  any  time  is  equal  to  the  product  of  velocity 
and  the  time  ;  and,  from  the  second,  we  see  that  the  velo- 
city is  equal  to  the  space  described  in  any  time,  divided  by 
that  time. 

These  laws  hold  true  for  all  cases  of  uniform  motion.  If 
we  denote  by  ds  the  space  described  in  the  infinitely  short 
time  dt,  we  shall  have,  from  the  last  principle, 

-S <»•> 


144  MECHANICS. 

which  is  the  differential  equation  of  uniform  motion,  v  being 
constant.  Clearing  this  equation  of  fractions,  and  integ- 
rating, we  have, 

8  =  vt  +  C      .     .     .     .     ( 57.) 

which  is  the  most  general  equation  of  uniform  motion.  If, 
in  (57),  we  make  t  =  0,  we  shall  have, 

s  =  O. 

Hence,  we  see  that  the  constant  of  integration  represent? 
the  space  passed  over  by  the  point,  from  the  origin  of  spaces 
up  to  the  beginning  of  the  time  t.  This  space  is  called  the 
initial  space.     Denoting  it  by  s\  we  have, 

s  =  vt  +  s'        .     .     .     .     (58.) 

If  s'  =  0,  the  origin  of  spaces  corresponds  to  the  origin 
of  times,  and  we  have, 

s  =  vt, 

the  same  as  the  first  of  Equations  (55.) 

Varied  Motion. 

113.  Varied  motion  is  that  in  which  the  velocity  is 
continually  changing.  It  can  only  result  from  the  action 
of  an  incessant  force. 

To  find  the  differential  equations  of  varied  motion,  let  us 
denote  the  velocity  at  the  time  t,  by  w,  and  the  space 
passed  over  up  to  that  time,  by  8.  In  the  succeeding  instant 
dt,  the  space  described  will  be  cfe,  and  the  velocity  gener- 
ated will  be  do.  Now,  the  space  <;&,  which  is  described  in 
the  infinitely  small  time  dt,  may  be  regarded  as  having  been 
described  with  the  uniform  velocity  v.  Hence,  from  Equa- 
tion (55),  we  have, 

»  =  S (59-> 

Let  us  denote  the  acceleration  due  to  the  incessant  force 
at  the  time  ty  by  <p.     We  have  seen  (Art.  24),  that  the  meas- 


RECTILINEAR    MOTION.  145 

ure  of  the  acceleration  due  to  a  force,  is  the  velocity  that  it 
can  impart  in  a  unit  of  time,  on  the  hypothesis  that  it  acts 
uniformly  during  that  time.  Now,  it  is  plain  that,  so  long 
as  the  force  acts  uniformly,  the  velocity  generated  will  be 
proportional  to  the  time,  and,  consequently,  the  measure  of 
the  acceleration  will  be,  the  quotient  obtained  by  dividing 
the  velocity  generated  in  any  time,  by  that  time.  The  quan- 
tity 9  is,  in  general,  variable ;  but  it  may  be  regarded  as 
constant  during  the  instant  dt ;  and  from  what  has  just  been 
said,  we  shall  have, 

•-a w 

Differentiating  Equation    (59),  we  have, 

d*s 
dv  =  -r-  ; 
dt  ' 

which,  being  substituted  in  Equation  (60)  gives, 

•=£ («•) 

Equations  (59),  (60),  and  (61)  are  the  differential  equa- 
tions required.  The  acceleration  <p  ,  is  the  measure  of  the 
force  exerted  when  the  mass  moved  is  the  unit  of  mass 
(Art.  24)  ;  in  any  other  case,  it  must  be  multiplied  by  the 
mass.  Denoting  the  entire  moving  force  applied  to  the 
mass  m  by  F,  we  shall  have, 

d^s 
F  =  mtp  =  m-jj     .     .     .     .     ( 62.) 
at 

This  value  of  F  is  the  measure  of  the  effective  moving 
force  in  the  direction  of  the  body's  motion.  When  a  body 
moves  upon  any  curve  in  space,  the  motion  may  be  regard- 
ed as  taking  place  in  the  direction  of  three  rectangular  axes. 
If  we  denote  the  effective  components  of  the  moving  force 
in  the  direction  of  these  axes,  by  Jf,  T",  and  Z,  the  spaces 
7 


146  MECHANICS. 

described  being  denoted  by  x,  y,  and  z,  we  shall  have,  from 

(62), 

^  d*x         „  cPy       „  cVz 


Uniformly  Varied  Motion. 

1 14.  Uniformly  varied  motion  is  that  in  which  the 
velocity  increases  or  diminishes  uniformly.  In  the  former 
case,  the  motion  is  accelerated  y  in  the  latter  case,  it  is  re- 
tarded. In  both  cases,  the  moving  force  is  constant.  De- 
noting the  acceleration  due  to  this  constant  force,  by/*,  we 
shall  have,  from  Equation  (61), 

v  =/ (») 

Multiplying  by  dt,  and  integrating,  we  have, 

%  =  ft+C     .     .     .     (64.) 

ds 

or,  since  —   is  equal  to   v,   Equation  (59), 
at 

V=ft+  G       ...     (65.) 


B". 


Multiplying  both  members  of  (64)  by  dt,  and  integratin 
we  have, 

8==if(*  +  Ct+  O       ...     (66.) 

Equations  (65)  and  (66)  express  the  relations  between 
the  velocity,  space,  and  time,  in  the  most  general  case  of 
uniformly  varied  motion.  These  equations  involve  the  two 
constants  of  integration  G  and  C,  which  serve  to  make 
them  conform  to  the  different  cases  that  may  arise.  To  de- 
termine the  value  of  these  constants,  make  t  —  0  in  the 
two  equations,  and  denote  the  corresponding  values  of  v 
and  s,  by  v'  and  s'.     We  shall  have, 

C  =t>'. 

C   =  8'. 


RECTILINEAR    MOTION.  14  J 

That  is,  C  is  equal  to  the  Telocity  at  the  beginning  of  the 
time  t,  and  C  is  equal  to  space  passed  over  up  to  the  same 
time.  These  values  of  the  velocity  and  space  are  called, 
respectively,  the  initial  velocity,  and  the  initial  space. 
Substituting  for  G  and  C  these  values  in  (65)  and  (66), 
they  become, 

v  =  v'  +ft (67.) 

s  =  sf  +  v't  +  \ftf     .     .     ( 68.) 

From  these  equations,  we  see  that  the  velocity  at  any 
eiine  t,  is  made  up  of  two  parts,  the  initial  velocity,  and  the 
velocity  generated  during  the  time  t  ;  we  also  see,  that  the 
space  is  made  up  of  three  parts,  the  initial  space,  the  space 
due  to  the  initial  velocity  for  the  time  t,  and  the  space  due 
to  the  action  of  the  incessant  force  during  the  same  time. 

By  giving  suitable  values  to  v'  and  s' ',  Equations  (67)  and 
(68)  may  be  made  to  express  every  phenomenon  of  varied 
motion.  If  we  suppose  both  v'  and  s'  equal  to  0,  the  body 
will  move  from  a  state  of  rest  at  the  origin  of  times,  and 
Equations  (67)  and  (68)  will  become, 

v  =  ft (69.) 

s  =  ±ft2 (70.) 

From  the  first  of  these  equations,  we  see  that,  in  uniformly 
varied  motion,  the  velocity  varies  as  the  time ;  and,  from 
the  second  one,  we  see  that  the  space  described  varies  as 
the  square  of  the  time. 

If,  in  Equation  (70),  we  make  t  =  1,  we  have, 


/;      or,   /  =  2*. 


That  is,  when  a  body  moves  from  a  state  of  rest,  under 
the  action  of  a  constant  force,  the  acceleration  is  equal  tc 
twice  the  space  passed  over  in  the  first  second  of  time. 

If,  in  the  preceding  equations,  we  suppose  f  to  be  essen- 
tially positive,  the  motion  will  be  uniformly  accelerated ;  if 
we  suppose  it  to  be  negative,  the  motion  will  be  uniformly 


148  MECHANICS. 

retarded.     In    the    latter   case,  Equations  (67)    and    (68) 

become, 

v  =  v'-ft (71.) 

8  =  8'  +  v't  -  \f?        .     .     .      ( 72.) 

Application  to  Falling  Bodies. 

115.  The  force  of  gravity  is  the  force  exerted  by  the 
earth  upon  all  bodies  exterior  to  it,  tending  to  draw  them 
towards  it.  It  is  found  by  observation,  that  this  force  is 
directed  towards  the  centre  of  the  earth,  and  that  its  intensity 
varies  inversely,  as  the  square  of  the  distance  from  the  centre. 

Since  the  centre  of  the  earth  is  so  far  distant  from  the 
surface,  the  variation  in  intensity  for  small  elevations  above 
the  surface  will  be  inappreciable.  Hence,  we  may  re- 
gard the  force  of  gravity  at  any  place  on  the  earth's  sur- 
face, and  for  small  elevations  at  that  place,  as  constant,  in 
which  case,  the  equations  of  the  preceding  article  become 
immediately  applicable.  The  force  of  gravity  acts  equally 
upon  all  the  particles  of  a  body,  and  were  there  no  resistance 
offered,  it  would  impart  the  same  velocity,  in  the  same  time, 
to  any  two  bodies  whatever.  The  atmosphere  is  a  cause  of 
resistance,  tending  to  retard  the  motion  of  all  bodies  fulling 
through  it ;  and  of  two  bodies  of  equal  mass,  it  retards  that 
one  the  most,  which  offers  the  greatest  surface  to  the  direc- 
tion of  the  motion.  In  discussing  the  laws  of  falling  bodies, 
it  will,  therefore,  be  found  convenient,  in  the  first  place,  to 
regard  them  as  being  situated  in  vacuum,  after  which,  a 
method  will  be  pointed  out,  by  means  of  which  the  veloci- 
ties can  be  so  diminished,  that  atmospheric  resistance  may 
be  neglected. 

Let  us  denote  the  acceleration  due  to  gravity,  at  any 
point  on  the  earth's  surface,  by  g,  and  the  space  fallen 
through  in  the  time  t,  by  h.  Then,  if  the  body  moves  from 
*  state  of  rest  at  the  origin  of  times,  Equations  (69)  and 
(70)  will  give, 

v  =  gt (  73.) 

h  =\g? (74.) 


KECTILTNEAR    MOTION. 


149 


From  these  equations,  we  see  that  the  velocities  at  two 
different  times  are  proportional  to  the  times,  and  the  spaces 
to  the  squares  of  the  times. 

It  has  been  found  by  experiment  that  the  velocity  im- 
parted to  a  body  in  one  second  of  time  by  the  action  of  the 
force  of  gravity  in  the  latitude  of  New  York,  is  about  32i 
feet.  Making  g  =  321  ft.5  and  giving  to  t  the  successive 
values  Is,  2s,  3s,  &c,  in  Equations  (73)  and  (74),  we  shall 
have  the  results  indicated  in  the  following 

TABLE. 


TIME    F.LAPSED. 

VELOCITIES    ACQUIRED. 

SPACES    DESCRIBED. 

SECONDS. 

FEET. 

FEET. 

1 

321 

16TV 

2 

64i 

641 

3 

961 

144-2 

4 

128J 

2571 

5 

150f 

402Ti5 

&C. 

&c. 

&c. 

Solving  Equation  (74)  with  respect  to  t,  we  have, 


t  = 


(75.) 


That  is,  the  time  required  for  a  body  to  fall  through  any 
height  is  equal  to  the  square  root  of  the  quotient  obtained 
by  dividing  twice  the  height  in  feet  by  32i. 

Substituting  this  value  of  t  in  Equation  (  73),  we  have. 


=  9* 


or    v 


9 


2ghi 


150  MECHANICS. 

whence,  by  solving  with  reference  to  v  and  h  respectively, 

v  =  V^gh,     and     h  =  —       .     .     ( 76.) 

These  equations  are  of  frequent  use  in  dynamical  investiga- 
tions. In  them  the  quantity  v  is  called  the  velocity  due  to  the 
height  A,  and  the  quantity  A,  the  height  due  to  the  velocity  v. 

If  we  suppose  the  body  to  be  projected  downwards  with 
a  velocity  v',  the  circumstances  of  motion  will  be  made 
known  by  the  Equations, 

v  =  v'  +  gt, 

h  =  v't  +  \gt\ 

In  these  equations  we  have  supposed  the  origin  of  spaces 
to  be  at  the  point  at  which  the  body  is  projected  down- 
wards. 

Motion  of  Bodies  projected  vertically  upwards. 

116.  Suppose  a  body  to  be  projected  vertically  upwards 
from  the  origin  of  spaces  with  a  velocity  v\  and  afterwards 
to  be  acted  upon  by  the  force  of  gravity.  In  this  case,  the 
force  of  gravity  acts  to  retard  the  motion.  Making  in  (71) 
and  (72),  s'  =  o,    f  =  g,    and    s  =  h,  they  become, 

v  —  v'  —  gt (77.) 

h  =  v't  —  \g?     .     .     .     .     ( 78.) 

In  these  equations,  h  is  positive  when  estimated  upwards 
from  the  origin  of  spaces,  and  consequently  negative,  when 
estimated  downwards  from  the  same  point. 

From  Equation  (77),  we  see  that  the  velocity  diminishes 
as  the  time  increases.     The  velocity  will  be  0,  when, 

v    —  gt  —  0,     or  when  t  =      • 

v' 
If  t  continues  to  increase  beyond  the  value   — ,    v  will 


KIX TIL.NKAR    MOTION.  151 

become  negative,  and  the  body  will  retrace  its  path.  Hence, 
the  time  required  for  the  body  to  reach  its  highest  elevation, 
is  equal  to  the  initial  velocity  divided  by  the  force  of 
gravity. 

Eliminating  t  from  Equations  (77)  and  (78),  we  have, 

v'*  —  ?r 

*=H- (V9>) 

Making   v  =  0,  in  the  last  equation,  we  have, 

h=   I- (80.) 

2g  v 

Hence,  the  greatest  height  to  which  the  body  will  ascend, 
is  equal  to  the  square  of  the  initial  velocity,  divided  by 
twice  the  force  of  gravity. 

This  height  is  that  due  to  the  initial  velocity  (Art.  115). 

v' 

If,  in  Equation  (77),  we  make   t  — t\   we  find, 

«/ 

V  =  9? (81.) 

v' 

If,  in  the  same  equation,  we  make    t  = \-  t\  we  find, 

t/ 

v=   -g? (82.) 

Hence,  the  velocities  at  equal  times  before  and  after 
reaching  the  highest  points,  are  equal. 

The  difference  of  signs  shows  that  the  body  is  moving  in 
opposite  directions  at  the  times  considered. 

If  we  substitute  these  values  of  v  success:  vely,  in  Equa- 
tion (£9),  we  shall,  in  both  cases,  find 

5"  -  gH'* 

~g       - 

which  shows  that  the  points  at  which  the  velocities  are 
equal,  both  in  ascending  and  descending,  are  equally  distant 
from  the  highest  point ;  that  is,  they  arc  coincident.     Hence, 


152  MECHANICS. 

if  a  body  be  projected  vertically  upwards,  it  will  ascend  to  a 
certain  point,  and  then  return  upon  its  path,  in  such  a  man* 
?ier,  that  the  velocities  in  ascending  and  descending  icill  be 
equal  at  the  same  points. 

EXAMPLES. 

1.  Through  what  distance  will  a  body  fall  from  a  state 
of  rest  in  vacuum,  in  10  seconds,  and  through  what  space  will 
it  fall  during  the  last  second  ?      Ans.   16081  ft.,  and  305^  ft. 

2.  In  what  time  will  a  body  fall  from  a  state  of  rest 
through  a  distance  of  1200  feet  ?  Ans.  8.63  sec. 

3.  A  body  was  observed  to  fall  through  a  height  of 
100  feet  in  the  last  second.  How  long  was  the  body  falling, 
and  through  what  distance  did  it  descend  ? 

SOLUTION. 

If  we  denote  the  distance  by  A,  and  the  time  by  t,  we 
shall  have, 

h  =  \gt\  and  h  -  100  =  Uj(t  -  l)2  ; 

.*.    t  =  3.6  sec,     and       A  —  208.44  ft.  Ans. 

4.  A  body  falls  through  a  height  of  300  feet.  Through 
what  distance  does  it  fall  in  the  last  two  seconds  ? 

The  entire  time  occupied,  is  4.32  sec.  The  distance  fallen 
through  in  2.32  sec,  is  86.57  ft.  Hence,  the  distance  re- 
quired is  300  ft.  —  86.57  ft.  =  213.43  ft.  Ans, 

5.  A  body  is  projected  vertically  upwards,  with  a  veloci- 
ty of  60  feet.     To  what  height  will  it  rise  ?        Ans.  55.9  ft. 

6.  A  body  is  projected  vertically  upwards  with  a  veloci- 
ty of  483  ft.  In  what  time  will  it  rise  to  a  height  of 
1610  feet  ? 

We  have,  from  Equation  (78), 

1610  =  483*  -  16^  ;     •••     t  =  Vvnr   ±  VoV  i 
or,  t  =  26.2  sec,  and  t  —  3.82  sec. 
The  smaller  value  of  t  gives  the  time  required  ;  the  larger 


RFXTILINEAR    MOTION.  153 

value  of  t  gives  the  time  occupied  in  rising  to  its  greatest 
height,  and  returning  to  the  point  which  is  1610  feet  from 
the  starting  point. 

V.  A  body  is  projected  vertically  upwards,  with  a  veloci- 
ty of  161  feet,  from  a  point  214J  feet  above  the  earth.  In 
what  time  will  it  reach  the  surface  of  the  earth,  and  with 
what  velocity  will  it  strike  ? 

SOLUTION. 

The  body  will  rise  from  the  starting  point  402.9  ft.  The 
time  of  rising  will  be  5  sec.  ;  the  time  of  falling  from  the 
highest  point  to  the  earth  will  be  6.2  sec.  Hence,  the  re- 
quired time  is  11.2  sec.     The  required  velocity  is  199  ft. 

8.  Suppose  a  body  to  have  fallen  through  50  feet,  when 
a  second  begins  to  fall  just  100  feet  below  it.  How  far  will 
the  latter  body  fall  before  it  is  overtaken  by  the  former  ? 

A  ns.  50  feet 
Restrained  Vertical  Motion. 

117.  We  have  seen  that  the  entire  force  exerted  in 
moving  a  body  is  equal  to  the  acceleration,  multiplied  by  the 
mass  (Art.  24).  Hence,  the  acceleration  is  equal  to  the 
moving  force,  divided  by  the  mass.  In  the  case  of  a  filling 
body,  the  moving  force  varies  directly  as  the  mass  moved  ; 
and,  consequently,  the  acceleration  is  independent  of  the 
mass.  If,  by  any  combination,  the  moving  force  can  be 
diminished  whilst  the  mass  remains  unchanged,  there  will  be 
a  corresponding  diminution  in  the  acceleration.  This  object 
may  be  obtained  by  the  combination  represented  in  the 
figure.  A  represents  a  fixed  pulley,  mounted 
on  a  horizontal  axis,  in  such  a  manner  that  the  /TX 
friction  shall  be  as  small  as  possible ;  W  and 
W  are  unequal  weights,  attached  to  a  flexible 
cord  passing  over  the  pulley.  If  we  suppose 
the  weight  IFgreater  than  W\  the  former  will  * 
descend  and  draw  the  latter  up.  If  the  dif- 
ference is  very  small,  the  motion  will  be  very  Fig.  99. 
slow,  and  if  the  instrument  is  nicely  constructed, 


154  MECHANICS. 

we  may  neglect  all  hurtful  resistances  as  inap- 
preciable.    Denote  the  masses  of  the  weights       ^_^^ 
W  and  IF',  by  m  and  m\  and  the  force  of      f  -A-  \ 
gravity,  by  g.     The  weight  W  is  urged  down- 
wards by  the  moving  force  mg,  and  this  mo- 
tion  is    resisted    by   the    moving   force   m'g. 
Hence,    the   entire   moving  force   is  equal  to     3W*    rw 
mg  —  m'g,  or,  (m  —  m')g,  and  the  entire  mass         P5t,  m 
moved,  is  m  -f-  m\  since  the  cord  joining  the 
weights  is  supposed  inextensible.     If  we  denote  the  accel 
eration  by  g',  we  shall  have,  from  what  was  said  at  the 
beginning  of  this  article, 

a'  = ,g (83.) 

J         m+  m! J  v       } 

By  diminishing  the  difference  between  m  and  m\  we  may 
make  the  acceleration  as  small  as  we  please.  It  is  plain  that 
g'  is  constant;  hence,  the  motion  of  TFis  uniformly  varied. 

wy~l    77c 

If  we  replace  g  by  — — — }g,  in  Equations  (73)  and   (74), 

711/  ~\~  771 

they  will  make  known  the  circumstances  of  motion  of  the 
body  W.  This  principle  is  employed  to  illustrate  the  laws 
of  falling  bodies  by  means  of  Atwood's  machine. 

Had  the  two  weights  under  consideration  been  attached 
to  the  extremities  of  cords  passing  around  a  wheel  and  its 
axle,  and  in  different  directions,  it  might  have  been  shown 
that  the  motion  would  be  uniformly  varied,  when  the  mo- 
ment of  either  weight  exceeded  that  of  the  other.  The 
same  principle  holds  good  in  the  more  complex  combinations 
of  pulleys,  wheels  and  axles,  &c.  In  practice,  however,  the 
hurtful  resistances  increase  so  rapidly,  that  even  when  the 
moving  force  remains  constant,  the  velocity  soon  attains 
a  maximum  limit,  after  which  the  motion  will  be  sensibly 

unifo-  ,n. 

j;  x  a  M  P  l  k  s. 

].  Two  weights  of  5  lbs.  and  4  lbs.,  respectively,  are 
suspended  from  the  extremities  of  a  cord  passing  over  a 


RECTILINEAR    MOTION.  155 

fixed  pulley.  What  distance  will  each  weight  describe  in  the 
tirst  second  of  time,  what  velocity  will  be  generated  in  one 
second,  and  what  will  be  the  tension  of  the  connecting  cord  ? 

SOLUTION. 

Since  the  masses  are  proportional  to  the  weights,  we 
shall  have, 

q'  =  i— ^  q  =  -    X   32-  ft.  =  3.574  ft. 
J  5  +  4J        9  6 

Hence,  the  velocity  generated  is  3.574  ft.,  and  the  space 
passed  over  is  1.787  ft.  To  find  the  tension  of  the  string, 
denote  it  by  x.  The  moving  force  acting  upon  the  heavier 
body,  is  (5  —  x)g,  and  the  acceleration  due  to  this  force, 

( — ^-)g\  the  moving  force  acting  upon  the  lighter  body, 

/rvt    ^.\ 

is  (x— 4)g,   and  the  corresponding  acceleration,  ( — - — W. 

But  since  the  two  bodies  move  together,  these  accelerations 
must  be  equal.     Hence, 

/5  —  x\  (X  —  4\ 

(— >  =  (-r>; 

.-.     x  —  4j  lbs.,  the  required  tension. 

2.  A  weight  of  1  lb.,  hanging  on  a  pulley,  descends  and 
drags  a  second  weight  of  5  lbs.  along  a  horizontal  plane. 
Neglecting  hurtful  resistances,  to  what  will  the  accelerating 
force  be  equal,  and  through  what  spr.ee  will  the  descending 
body  move  in  the  first  second  ? 

solution. 
The  moving  force  is  equal  to  1  x  </,  and  the  mass  moved 
is  equal  to  6.    Hence,  the  acceleration  is  equal  to  -   —  5.3622 
ft.,  and  the  space  described  will  be  equal  to  2.6811  ft. 

3.  Two  bodies,  each  weighing  5  lbs.,  are  attached  to  a 
string  passing  over  a  fixed  pulley.     What  distance  will  each 


io6 


MECHANICS. 


body  move  in  10  seconds,  when  a  pound  weight  is  added  to 
one  of  them,  and  what  velocity  will  have  been  generated  at 
the  end  of  that  time  ? 

SOLUTION. 

The  acceleration  will  be  equal  to  T\g  =  2.924  ft.  =  gr. 


But,  s  =  \g't\ 


g't      Hence,  the  space  described  in  10 


seconds  is  146.2  ft.,  and  the  velocity  generated  is  29.24  ft. 

4.  Two  weights,  of  16  oz.  each,  are  attached  to  the  ends 
of  a  string  passing  over  a  fixed  pulley.  What  weight  must 
be  added  to  one  of  them,  that  it  may  descend  through  a 
foot  in  two  seconds  ? 

SOLUTION. 

Denote  the   required  weight  by  x ;  the  acceleration  will 


be  equal  to 


g  =  g'.  ^But  8  —  ^-g't- :  making  s  =  1 


X  324 


0.505  oz.  Ans. 


32  +  x 
and   t  =  2,    we  have, 

2x 
32  +  x 

Atwood's  Machine. 
118.  Atwood's  machine  is  a  contrivance  to  illustrate  the 
laws  of  falling  bodies.  It  consists  of  a  vertical 
post  AB,  about  12  feet  in  height,  supporting, 
at  its  upper  extremity,  a  fixed  pulley  A.  To 
obviate,  as  much  as  possible,  the  resistance  of 
friction,  the  axle  is  made  to  turn  upon  friction 
rollers.  A  fine  silk  string  passes  over  the 
pulley,  and  at  its  two  extremities  are  fastened 
two  equal  weights  (J  and  D.  In  order  to 
impart  motion  to  the  weights,  a  small  weight 
6r,  in  the  form  of  a  bar,  is  laid  upon  the 
weight  (7,  and  by  diminishing  its  mass,  the 
acceleration  may  be  rendered  as  small  as 
desirable.  The  vertical  rod  AB,  graduated 
to  feet  and  decimals,  is  provided  with  two 
sliding  stages  E  and  E\  the  upper  one  is  in 


the  form   of  a   ring,  which  will  permit  the 


Fig.  100. 


RECTILINEAR    MOTION.  157 

weight  (7,  to  pass,  but  not  the  bar  G ;  the  lower  one  is  in 
the  form  of  a  plate,  which  is  intended  to  intercept  the 
weight  G.  There  is  also  connected  with  the  instrument  a 
seconds  pendulum  for  measuring  time. 

Let  us  suppose  that  the  weights  of  C  and  7>,  are  each 
equal  to  181  grains,  and  that  the  weight  of  the  bar  G,  is 
24  grains.     Then  will  the  acceleration  be 

q'  =  2- g  =  2  ft. ; 

y  362  +  24  J  ' 

and  since  h  =  ^g'f,  and  v  =  g't  (Art.  115),  we  shall 
have,  for  the  case  in  question, 

h  =  t\    and     v  =  2t. 

If,  in  these  equations,  we  make  t  =.  1  sec,  we  shall 
have  h  =  1,  and  v  =  2.  If  we  make  t  =  2  sec,  we  shall, 
in  like  manner,  have  h  =  4,  and  «  =  4.  If  we  make 
t  =  3  sec,  we  shall  have  A  =  9,  and  v  =  6,  and  so  on. 
To  verify  these  results  experimentally,  commencing  with 
the  first.  The  weight  C  is  drawn  up  till  it  comes  opposite 
the  0  of  the  graduated  scale,  and  the  bar  G  is  placed  upon 
it.  The  weight  thus  set  is  held  in  its  place  by  a  spring. 
The  ring  E  is  set  at  1  foot  from  the  0,  and  the  stage  JF]  is 
set  at  3  feet  from  the  0.  When  the  pendulum  reaches  one 
of  its  extreme  limits,  the  spring  is  pressed  back,  the  weight 
(7,  G  descends,  and  as  the  pendulum  completes  its  vibration, 
the  bar  G  strikes  the  ring,  and  is  retained.  The  acceleration 
then  becomes  0,  and  the  weight  C  moves  on  uniformly,  with 
the  velocity  that  it  had  acquired,  in  the  first  second  ;  and  it 
will  be  observed  that  the  weight  C  strikes  the  second  stage 
just  as  the  pendulum  completes  its  second  vibration.  Had 
the  stage  F  been  st1  at  5  feet  from  the  0,  the  weight  0 
would  have  reached  it  at  the  end  of  the  third  vibration  of 
the  pendulum.  Had  it  been  7  feet  from  the  0,  it  would 
have  reached  it  at  the  end  of  the  fourth  vibration,  and  so  on. 

To  verify  the  next  result,  we  set  the  ring  E  at  four  feet 


158  MECHANICS. 

from  the  0,  and  the  stage  F  at  8  feet  from  the  0,  and  pro- 
ceed as  before.  The  ring  will  intercept  the  bar  at  the  end 
of  the  first  vibration,  and  the  weight  will  strike  the  stage  at 
the  end  of  the  second  vibration,  and  so  on. 

By  making  the  weight  of  the  bar  less  tnan  24  grains,  the 
acceleration  is  diminished,  and,  consequently,  the  spaces  and 
velocities  correspondingly  diminished.  The  results  may  be 
verified  as  before. 

Motion  of  Bodies  on  Inclined  Planes. 

119.  If  a  body  be  placed  on  an  inclined  plane,  and 
abandoned  to  the  action  of  its  own  weight,  it  will  either 
slide  or  roll  down  the  plane,  provided  there  be  no  friction 
between  it  and  the  plane.  If  the  body  is  spherical,  it  will 
roll,  and  in  this  case  the  friction  may  be  disregarded.  Let 
the  weight  of  the  body  be  resolved  into  two  components ; 
one  perpendicular  to  the  plane,  and  the  other  parallel  to  it. 
The  plane  of  these  components  will  be  vertical,  and  it  will 
also  be  perpendicular  to  the  given  plane.  The  effect  of  the 
first  component  will  be  counteracted  by  the  resistance  of  the 
plane,  whilst  the  second  component  will  act  as  a  constant 
force,  continually  urging  the  body  down  the  plane.  The 
force  beins  constant,  the  body  will  have  a  uniformly  varied 
motion,  and  Equations  (67)  and  (08)  will  be  immediately 
applicable.  The  acceleration  will  be  found  by  projecting 
the  acceleration  due  to  gravity  upon  the  inclined  plane. 

Let  AP  represent  a  section  of  the  inclined  plane  made  by 
a  vertical  plane  taken  perpendicular 
to  the  given  plane,  and  let  P  be  the 
centre  of  gravity  of  a  body  resting 
on  the  given  plane.  Let  PQ  repre- 
sent the  acceleration  due  to  gravity, 
denoted   by  #,  and  let  PR  be  the  Fig.  101. 

component  of  y,  which  is  parallel  to 

AB,  denoted  by  ff\  PS  being  the  normal  component. 
Denote  the  angle  that  AB  makes  with  the  horizontal  plane 
by  a.     Then,  since  PQ  is  perpendicular  to  PC,  and  QP  to 


RECTILINEAR    MOTION.  159 

AB,  the  angle  BQP  is  equal  to  ABC,  or  to  a.     Hence  we 
have,  from  the  right-angled  triangle  BQB, 

g'  =  gmna. 

But  the  triangle  AB  C  is  right-angled,  and,  if  we  denote 
its  height  A  C  by  h,  and  its  length  AB  by  I,  we  shall  have 

sina  =  —  ,  which,  being  substituted  above,  gives, 

*'  =  f (84.) 

This  value  of  g'  is  the  value  of  the  acceleration  due  to  the 
moving  force.  Substituting  it  for  f  in  Equations  (67)  and 
(68),  we  have, 

s  =  S'  +  v't  +  (fle- 

If  the  body  starts  from  rest  at  A,  taken  as  the  origin  of 
spaces,  then  will  v'  =  0    and  -s'  =  0,  giving, 

V  =  ft (85.) 

•-■£'■  •  • (««•) 

To  find  the  time  required  for  a  body  to  move  from  the 
top  to  the  bottom  of  the  plane,  make  8  =■  I,  in  (86)  ;  there 
will  result, 

i --fit  ,      ..    i-y^S  *  (87° 

Hence  the  time  varies  directly  as  the  length,  and  inversely 
as  the  square  root  of  the  height. 

For  two  planes  having  the  same  height,  but  different 
lengths,  the  radical  factor  of  the  value  of  t  will  remain  con- 


160  MECHANICS. 

stant.  Hence,  tne  limes  required  for  a  body  to  move  down 
any  two  planes  having  the  same  height,  are  to  each  other  as 
their  lengths. 

To  determine  the  velocity  with  which  a  body  reaches  the 
bottom  of  the  plane,  substitute  for  t,  in  Equation  (85)  its 
value  taken  from  Equation  (86).  We  shall  have,  after 
reduction, 

v  =  ^/2gh. 

But  this  is  the  velocity  due  to  the  height  h  (Art.  115). 
Hence,  the  velocity  generated  in  a  body  whilst  moving 
down  any  inclined  plane,  is  equal  to  that  generated  in 
falling  freely  through  the  height  of  the  plane. 

EXAMPLES. 

1.  An  inclined  plane  is  10  feet  long  and  1  foot  high. 
How  long  will  it  take  for  a  body  to  move  from  the  top  to 
the  bottom,  and  what  velocity  will  it  acquire  in  the 
descent  ? 

SOLUTION. 

We  have,  from  Equation  (8V), 


t  =  lx. 

gh 

substituting  for  I  its  value  10,  and  for  h  its  value  1,  we  have, 
t  —  2\  seconds  nearly. 

From  the  formula  v  —  \/2gh,  we  have,  by  making 
h  =  1, 

V  =  -v/64.33  =  8.02  ft. 

2.  How  far  will  a  body  descend  from  rest  in  4  seconds, 
on  an  inclined  plane  whose  length  is  400  feet,  and  whose 
height  is  300  fee*  ?  Ans.  103  t't. 

3.  How  long  will  it  take  for  a  body  to  descend  100  feet 
on  a  plane  whose  length  is  150  feet,  and  whose  height  is  00 
feet?  Ans.  3.9  sec 


KECTILINEAR    MOTION.  161 

4.  There  is  an  inclined  railroad  track,  2£  miles  in  length, 
whose  inclination  is  1  in  35.  What  velocity  will  a  car 
attain,  in  running  the  whole  length  of  the  road,  by  its  own 
weight,  hurtful  resistances  being  neglected  ? 

Am.     155.75  ft.,  or,  106.2  m.  per  hour. 

5.  A  railway  train,  having  a  velocity  of  45  miles  per 
hour,  is  detached  from  the  locomotive  on  an  ascending  grade 
of  1  in  200.  How  far,  and  for  what  time,  will  the  train 
continue  to  ascend  the  inclined  plane  ? 

SOLUTION. 

We  find  the  velocity  to  be  66  ft.  per  second.  Hence, 
66  =  y/'lgh  ;  or,  h  =  67.7  ft.  for  the  vertical  height. 
Hence,  67.7  X  200  =  13,540  ft.,  or,  2.5644  m.,  the  distance 
which  the  train  will  proceed.     We  have, 

t  =  I  \I  — t  =  410.3  sec,    or,  6  min.  50.3  sec, 
foi  the  time  required  to  come  to  rest. 

6.  A  body  weighing  5  lbs.  descends  vertically,  and  draws 
a  weight  of  6  lbs.  up  an  inclined  plane  of  45°.  How  far 
will  the  first  body  descend  in  10  seconds  ? 

SOLUTION. 

The  moving  force  is  equal  to  5  —  6  sin  45°  ;  and,  conse. 
sequently,  the  acceleration, 

5-6sin45°        .757         ^0010 
j  =  ____  =  _  =  .068818; 

.-.     s  =  \g'f  =  3.4409  ft.      Arts. 

Motion  of  a  Body  down  a  succession  of  Inclined  Planes. 

120.  If  a  body  start  from  the  top  of  an  inclined  plane, 
wTith  an  initial  velocity  v\  it  will  reach  the  bottom  with  a 
velocity  equal  to  the  initial  velocity,  increased  by  that  due 
to  the  height  of  the  plane.  This  velocity,  called  the  terminal 
velocity,  will,  therefore,  be  equal  to  that  which  the  body 


162  MECHANICS. 

would  have  acquii  =d  by  falling  freely  throngh  a  height  equal 

to  that  due  to  the  initial  velocity,  increased  by  that  of  the 

plane.     Hence,   if  a  body  start  from 

a  state  of  rest  at  A,  and,  after  having 

passed  over  one  inclined  plane  AB, 

enters    upon    a    second   plane    B  C, 

without  loss  of  velocitv,  it  will  reach 

»  '  Fig.  102. 

the  bottom  of  the  second  plane  with 

the  same  velocity  that  it  would  have  acquired  by  falling 
freely  through  DC,  the  sum  of  the  heights  of  the  two 
planes.  Were  there  a  succession  of  inclined  planes,  so  ar- 
ranged that  there  would  be  no  loss  of  velocity  in  passing 
from  one  to  another,  it  might  be  shown,  by  a  similar  course 
of  reasoning,  that  the  terminal  velocity  would  be  equal  to 
that  due  to  the  vertical  distance  of  the  terminal  point  below 
the  point  of  starting. 

By  a  course  of  reasoning  entirely  analagous  to  that  em- 
ployed in  discussing  the  laws  of  motion  of  bodies  projected 
vertically  upwards,  it  might  be  shown  that,  if  a  body  were 
projected  upwards,  in  the  direction  of  the  lower  plane,  with 
the  terminal  velocity,  it  would  ascend  along  the  several 
planes  to  the  top  of  the  highest  one,  where  the  velocity 
would  be  reduced  to  0.  The  body  would  then,  under  the 
action  of  its  own  weight,  retrace  its  path  in  such  a  manner 
that  the  velocity  at  every  point  in  descending  would  be  the 
same  as  in  ascending,  but  in  a  contrary  direction.  The  time 
occupied  by  the  body  in  passing  over  any  part  of  its  path  in 
descending,  would  be  exactly  equal  to  that  occupied  in 
passing  over  the  same  portion  in  ascending. 

In  the  preceding  discussion,  we  have  supposed  that  there 
is  no  loss  of  velocity  in  passing  from  one  plane  to  another. 
To  ascertain  under  what  circumstances  this  condition  will  be 
fulfilled,  let  us  take  the  two  planes  AB  and  BC.  Prolong 
BG  upwards,  and  denote  the  angle  ABE,  by  9.  Denote 
the  velocity  of  the  1m><1v  on  reaching  B,  by  v'.  Let  v'  be 
resolved  into  two  components,  one  in  the  direction  of  BC, 
and  the  other  at  right  angles  to  it.     The  effect  of  the  latter 


TEKIODIC    MOTION.  163 

will  bo  destroyed  by  the  resistance  of  the  plane,  and  the 
former  will  be  the  effective  velocity  in  the  direction  of  the 
plane  B  C.  •  From  the  rule  for  decomposition  of  velocities, 
we  have,  for  the  effective  component  of  v\  the  value  v'  cos?. 
Hence,  the  loss  of  velocity  due  to  change  of  direction,  is 
v'  —  v'  cos$  ;  or,  v'{\  —  cos?),  which  is  equal  to  v'  ver-sinr. 
But  when  cp  is  infinitely  small,  its  versed-sine  is  0,  and  there 
will  be  no  loss  of  velocity.  Hence,  the  loss  of  velocity  due 
to  change  of  direction  will  always  be  0,  when  the  path  of 
the  body  is  a  curved  line.  This  principle  is  general,  and 
may  be  enunciated  as  follows  :  When  a  body  is  constrained 
to  describe  a  curvilinear  path,  there  will  be  no  loss  of  velo- 
city in  consequence  of  the  change  in  direction  of  the  body^s 
motion. 

Periodic  Motion.         t 

121.  Periodic  motion  is  a  kind  of  variable  motion,  in 
which  the  spaces  described  in  certain  equal  periods  of  time 
are  equal.  This  kind  of  motion  is  exemplified  in  the  pheno- 
mena of  vibration,  of  which  there  are  two  cases. 

1st.  Rectilinear  vibration.  Theory  indicates,  and  experi- 
ment confirms  the  fact,  that  if  a  particle  of  an  elastic  fluid 
be  slightly  disturbed  from  its  place  of  rest,  and  then  aban- 
doned, it  will  be  urged  back  by  a  force,  varying  directly  as 
its  distance  from  the  position  of  equilibrium  ;  on  reaching 
this  position,  the  particle  will,  by  virtue  of  its  inertia,  pass 
to  the  other  side,  again  to  be  urged  back,  and  so  on.  To 
determine  the  time  required  for  the  particle  to  pass  from 
one  extreme  position  to  the  opposite  one  and  back,  let  us 
denote  the  displacement  at  any  time  t  by  s,  and  the  accelera- 
tion due  to  the  restoring  force  by  <p  ;  then,  from  the  law  of 
the  force,  we  shall  have  9  =  rfs,  in  which  n  is  constant  for 
the  same  fluid  at  the  same  temperature.  Substituting  for  <p 
its  value,  Equation  (Gl),  and  recollecting  that  <p  acts  in  a 
direction  contrary  to  that  in  which  s  is  estimated,  we  have, 

<P* 

-a?=nS 


16-1  MECHANICS. 

Multiplying  both  members  by  2ds,  we  have, 
2dsdis 


dt>       ~2n'sds> 


whence,  by  integration, 


*  =  »v  +  o  =  -+. 


The   velocity  v  will  be  0  when   s  is  greatest   possible ; 
denoting  this  value  of  s  by  a,  we  shall  have, 

?iW  +(7=0;     whence,     C  =  -  n*a*. 

Substituting  this  value  of  G  in  the  preceding  equation,  it 
becomes, 

v2  =  -j-j  =  n2  (a2  —  s2)  ;    whence,    ndt  =  — -  .  (88.) 


dt2 


Integrating  the  last  equation,  we  have, 


Vaa  — 


?it+  C  =  sin-1  -     .     .         .     .     (89.) 
a 


Taking  the  integral  between  the  limits  s  =  -f-  a  and 
s  =  —  a,  and  denoting  the  corresponding  time  by  ±r 
t  being  the  time  of  a  double  vibration,  we  have, 

Inr  =  x  ;    whence,     r  =  —  • 

The  value  of  <r  is  independent  of  the  extent  of  the  excur- 
sion, and  dependent  only  upon  n.  Hence,  in  the  same 
medium,  and  at  the  same  temperature,  the  time  of  vibration 
is  constant. 

These  principles  are  of  utility  in  discussing  the  subjects 
of  sound,  light,  &c. 


PERIODIC    MOTION.  165 

2ndly.    Curvilinear  vibration.     Let  ABC  be  a  vertical 
plane    curve,    symmetrical  with 
respect  to  DB.     Let  AC  be  a 
horizontal  line,  and  denote  the 
distance  EB  by  h.     If  a  body  / 

■were  placed  at  A  and  abandoned        (£;.''_ 

to  the  action  of  its  own  weight,  cNc 

being  constrained  to  remain  on  ^^ 

the  curve,  it  would,  in  accord-  Fi    103 

ance  with  the  principles  of  the 

last  article,  move  towards  B  with  an  accelerated  motion, 
and,  on  arriving  at  B,  would  possess  a  velocity  due  to  the 
height  h.  By  virtue  of  its  inertia,  it  would  ascend  the 
branch  B  C  with  a  retarded  motion,  and  would  finally  reach 
(7,  where  its  velocity  would  be  0.  The  body  would  then  be 
in  the  same  condition  that  it  was  at  A,  and  would,  conse- 
quently, descend  to  B  and  again  ascend  to  A,  whence  it 
would  again  descend,  and  so  on.  TVere  there  no  retarding 
causes,  the  motion  would  continue  for  ever.  From  what 
has  preceded,  it  follows  that  the  time  occupied  by  the  body 
in  passing  from  A  to  B  is  equal  to  that  in  passing  from  B 
to  (7,  and  also  the  time  in  passing  from  C  to  B  is  equal  to 
that  in  passing  from  B  to  A.  Further,  the  velocities  of  the 
body  when  at  G  and  IT,  any  two  points  lying  on  the  same 
horizontal,  are  equal,  either  being  that  due  to  the  height 
EK.  These  principles  are  of  utility  in  discussing  the 
pendulum. 

Angular  Velocity. 

122.  "When  a  body  revolves  about  an  axis,  its  points 
being  at  different  distances  from  the  axis,  will  have  different 
velocities.  The  angular  velocity  is  the  velocity  of  a  point 
whose  distance  from  the  axis  is  equal  to  1.  To  obtain  the 
velocity  of  any  other  point,  we  multiply  its  distance  from 
the  axis  by  the  angular  velocity.  To  find  a  general  expres- 
sion for  the  velocity  of  any  point  of  a  revolving  body,  let  us 
denote  the  angular  velocity  by  w,  the  space  passed  over  by 
a  point  at  the  unit's  distance  from  the  axis  in  the  time  dt, 


166 


MECHANICS. 


by  c$.  The  quantity  cZA  is  an  infinitely  small  arc,  having  a 
radius  equal  to  1  ;  and,  as  in  Art.  113,  it  is  plain  that  we 
may  regard  the  angular  motion  as  uniform,  during  the  infin- 
itely small  time  dt.     Hence,  as  in  Article  113,  we  have, 


dd 
dt 


(90.) 


If  we  denote  the  distance  of  any  point  from  the  axis  by  I, 
and  its  velocity  by  v,  we  shall  have, 


v  =  flu  ;    or,  v 


l~ 
dt 


(91.) 


The  Simple  Pendulum. 

123.  A  pendulum  is  a  heavy  body  suspended  from  a 
horizontal  axis,  about  which  it  is  free  to  vibrate.  In  order 
to  investigate  the  circumstances  of  vibration,  let  us  first 
consider  the  hypothetical  case  of  a  single  material  point 
vibrating  about  an  axis,  to  which  it  is  attached  by  a  rod  des- 
titute of  weight.  Such  a  pendulum  is  called  a  simple  pen- 
dulum. The  laws  of  vibration,  in  this  case,  will  be  identical 
with  those  explained  in  Art.  121,  the  arc  ABC  being  the 
arc  of  a  circle.     The  motion  is,  therefore,  periodic. 

Let  ABC  be  the  arc  through 
which  the  vibration  takes  place,  and 
denote  its  radius  by  I.  The  angle 
CD  A  is  called  the  amplitude  of  vi- 
bration;  half  of  this  angle  ABB, 
denoted  by  a,  is  called  the  angle  of 
deviation  ;  and  I  is  called  the  length 
of  the  pendulum.  If  the  point  starts 
from  rest,  at  A,  it  will,   on  reaching 

any  point  JI,  of  its  path,  have  a  velocity  w,  due  to  the  height 
EK,  denoted  by  h.     Hence, 

v  =  \/2gh (92. 

If  we  denote  the  variable  angle  IIBB  by   6,   we  shall 


PERIODIC    MOTION.  167 

have   DK  =  Zcos3 ;    we  shall  also  have   DE  =  /cosa  ;  and 
since  h  is  equal  to  DK  —  DE,  we  shall  have, 

h  =  I  (cosd  —  cosa). 

Which,  being  substituted  in  the  preceding  formula,  gives, 


v  =  y2^(cos)  —  cosa). 

From  the  preceding  article,  we  have, 

,dd 
V=ldt' 

Equating  these  two  values  of  v,  we  have, 

d6 


I  -j ■  =   y2gl(cosd  —  cosa). 

Whence,  by  solving  with  respect  to  dt, 

fl  & 

dt  =  </—'    ,      |  .     .     .     (93.) 

V  2g  y  cos  J  —  cosa 

If  we  develop  cosJ  and  cosa  into  series,  by  McLaukin's 
theorem,  we  shall  have, 

00rf  =  l_|+_il__4fc&; 


C08a=l  __  +  ___   &c. 

WTien  a  is  very  small,  say  one  or  two  degrees,  d  being 
still  smaller,  we  may  neglect  all  the  terms  after  the  second 
as  inappreciable,  giving 


cosd  =  1  —  —; 


a* 

cosa  =  1 — : 

2 


168  MECHANICS. 

Or,  cos:)  —  cosa  —  ±(a2  —  d3 ). 

Substituting  in  Equation  (93),  it  becomes, 
1        M 


dt  = 

9 


(94.) 


Integrating  Equation  (94),  we  have, 

*  =  J1  sin-i  -  +  a 
V  a  a 


Taking  the  integral  between  the  limits  6  =  —  a,  and 
0  =  +  a,  t  will  denote  the  time  of  one  vibration,  and  we 
shall  have, 

fl 
t  z=  *  */-  (95.) 

V  g  v      ; 

Hence,  the  time  of  vibration  of  a  simple  pendulum  is 
equal  to  the  number  3.1416,  multiplied  into  the  square  root 
of  the  quotient  obtained  by  dividing  the  length  of  the  pen- 
dulum by  the  force  of  gravity. 

For  a  pendulum,  Avhose  length  is  V,  we  shall  have, 

t'  =  «  Kr- (96.) 

V  g 

From  Equations  (95)  and  (96),  we  have,  by  division, 

t         Vl  r-       - 

Y=~^fn     or,     t'.fuVf:  y/V  (97.) 

That  is,  the  times  of  vibration  of  two  simple  pendidums, 
are  to  each  other  as  the  square  roots  of  their  lengths. 

If  we  suppose  the  lengths  of  two  pendulums  to  be  the 
same,  but  the  force  of  gravity  to  vary,  as  it  does  slightly  in 
different  latitudes,  and  at  different  elevations,  we  shall  have, 

and     r=rX/i- 

9  V  9 


PERIODIC    MOTION.  169 

Whence,  by  division, 


V=v/->      or'      t:t"  ::  ^:   V9    -    (^) 
t  v    g 

That  is,  the  times  of  vibration  of  the  same  simple  pen- 
dulum, at  two  different  places,  are  to  each  other  inversely  as 
the  square  roots  of  the  forces  of  gravity  at  the  two  places. 

If  we  suppose  the  times  of  vibration  to  be  the  same,  and 
the  force  of  gravity  to  vary,  the  lengths  will  vary  also,  and 
we  shall  have, 

t  =  if  \  /  -  ,     and     t  =  if  \  /  —  • 
Equating  these  values  and  squaring,  we  have, 


I      v  t 

_              ,./    J 

or, 

1:1': 

:  9  '■ 

:  9' 

9      9 

(99.) 


That  is,  the  lengths  of  simple  pendulums  which  vibrate  in 
equal  times  at  different  places,  are  to  each  other  as  the 
forces  of  gravity  at  those  places. 

Vibrations  of  equal  duration  are  called  isochronal. 

The  Compound  Pendulum. 

124.  A  compound  pendulum  is  a  heavy  body  free  to 
oscillate  about  a  horizontal  axis.  This  axis  is  called  the 
axis  of  suspension.  The  straight  line  drawn  from  the 
centre  of  gravity  of  the  pendulum  perpendicular  to  the  axis 
of  suspension  is  called  the  axis  of  the  pendulum. 

In  all  practical  applications,  the  pendulum  is  so  taken  that 
the  plane  through  the  axis  of  suspension  and  the  centre  of 
gravity  divides  it  symmetrically. 

Were  the  elementary  particles  of  the  pendulum  entirely 
disconnected,  but  constrained  to  remain  at  invariable  dis- 
tances from  the  axis  of  suspension,  we  should  have  a  col- 
lection of  simple  pendulums.  Those  at  equal  distances  from 
8 


170  MECHANICS. 

the   axis  would  vibrate  in   equal   times;    those   unequally 
distant  from  it  would  vibrate  in  unequal  times. 

Those  particles  which  are  at  the  same  distance  from  the 
axis  of  suspension  lie  upon  the  surface  of  a  cylinder,  whose 
axis  coincides  with  the  axis  of  suspension,  and  we  may,  with- 
out at  all  affecting  the  time  of  vibration,  suppose  them  all  to 
be  concentrated  at  the  point  in  which  the  cylinder  cuts  the 
axis  of  the  pendulum.  If  we  suppose  the  same  to  be  done  for 
each  of  the  concentric  cylinders,  Ave  may  regard  the  pendu- 
lum as  made  up  of  a  succession  of  heavy  points,  a,  b,  .  .  .  p,  k, 
lying  on  the  axis,  firmly  connected  with  each 
other  and  with  the  point  of  suspension  C. 
The  particles  a,  b,  &c,  nearest  to  C  will  tend 
to  accelerate  the  motion  of  the  entire  pendu-  , 

lum,  whilst  those  most  remote,  as  p,  k,  <fcc,  / 

will  tend  to  retard  it.     There  must,  therefore,  / 

be  some  intermediate  point,  as  0,  which  will  h- 

vibrate  precisely  as  though  it  were  not   con-  F[cr  105 

nected  with  the  system ;  were  the  entire  mass 
of  the  pendulum  concentrated  at  this  point  it  would  vibrate 
in  the  same  time  as  the  given  pendulum.  This  point  0  is 
called  the  centre  of  oscillation.  Hence,  the  centre  of  oscil- 
lation of  a  pendulum  is  that  point  of  its  axis,  at  which,  if 
the  entire  mass  of  the  pendulum  were  concentrated,  its  time 
of  vibration  would  be  unchanged.  A  line  drawn  through 
this  point,  parallel  to  the  axis  of  suspension,  is  called  the  axis 
of  oscillation.  The  distance  from  the  axis  of  oscillation  to 
the  axis  of  suspension  is  the  length  of  an  equivalent  simple 
pendulum,  that  is,  of  a  simple  pendulum,  whose  time  of 
vibration  is  the  same  as  that  of  the  compound  pendulum. 

To  find  an  expression  for  CO,  C  being  the  axis  of  sus- 
pension, and  0  the  axis  of  oscillation.  Denote  CO  by  I; 
let  G  be  the  centre  of  gravity,  and  denote  the  distance 
CG  by  /•;  denote  the  masses  concentrated  at  a,  b,  .  .  .p,  k, 
by  m,  m'  .  .  .  m" ,  m'",  and  their  distances  from  C  by 
r,  r'  .  .  .  r",  r'". 

Whatever  may  be  the  position  of  CO,  the  pffective  com 


PERIODIC    MOTION.  171 

ponent  of  gravity  is  the  same  for  each  particle,  and  were 
they  free  to  move,  each  would  have  impressed  upon  it  the 
same  velocity  that  is  actually  impressed  upon  0.  Denote 
the  angular  velocity  at  any  instant,  by  w  ;  then  will  the 
actual  velocity  of  the  mass  m,  be  equal  to  poj,  and  the  effec- 
tive moving  force  will  be  equal  to  mru  (Art.  24).  Had  the 
mass  m  been  at  0,  instead  of  at  a,  the  entire  moving  force 
impressed  would  have  been  effective,  and  its  measure  would 
have  been  mlu.  The  difference  between  these  forces,  or 
m(l  —  r)w,  is  that  portion  of  the  force  applied  at  a  which 
goes  to  accelerate  the  motion  of  the  system.  The  moment 
of  this  force  with  respect  to  C,  is  m{l  —  r)ru.  In  like 
manner,  for  the  force  acting  at  &,  which  also  tends  to  accel- 
erate the  system,  we  have  m'{l  —  r,)r/«,  and  so  on,  for  all 
of  the  particles  between  0  and  C.  By  a  similar  course  of 
reasoning,  we  get,  for  the  moments  of  the  force  tending  to 
retard  the  system,  and  which  are  applied  at  the  points 
p,  k,  <fcc.,  m"(r"  -  l)r"u,  m"'{r'"  -  *)r'"co,  &c.  But  since 
there  is  neither  acceleration  nor  retardation,  in  consequence 
of  the  action  of  these  forces,  they  must  be  in  equilibrium, 
and,  consequently,  the  sum  of  the  moments  of  the  forces 
which  tend  to  accelerate  the  system,  must  be  equal  to  the 
sum  of  the  moments,  which  tend  to  retard  the  system 
Hence,  we  have, 

m(l  —  r)ru  +  m'(l  —  r')r'u  +  &c. 

-  m"(r"  _  i)r"u  +  m'"{r'"  -  l)r'"u  +  &c. 

Striking  out  the  factor  w,  and  reducing,  we  have, 
(rar  +  m'r'-\-m"r"  +  &c.)  I  =  mr*  +  mV"  +  m"r"2-\-  &c.% 
or,  2(mr)  X  I  =  ^(mr1). 

Hence, 

lsBS52  ....  (100.) 


172  MECHANICS. 

The  expression  2(mr'),  is  called  the  moment  of  inertia 
of  the  body  with  respect  to  the  axis  of  suspension. 

The  moment  of  inertia  of  a  body,  with  respect  to  any 
axis,  is  the  algebraic  sum  of  the  products  obtained  by  mul- 
tiplying the  mass  of  each  elementary  particle  by  tJte  square 
of  its  distance  from  the  axis. 

The  expression  I(mr),  is  called  the  moment  of  the  mass, 
with  respect  to  the  axis  of  suspension. 

Tlie  moment  of  the  mass  with  respect  to  any  axis,  is  the 
algebraic  sum  of  the  products  obtained  by  multiplying  the 
mass  of  each  elementary  particle  by  its  distance  from  the 
axis. 

From  the  principle  of  moments,  this  is  equal  to  the  mo- 
ment of  the  entire  mass,  concentrated  at  the  centre  of  grav- 
ity. Denote  the  mass,  or  2(m),  by  31,  the  distance  of  its 
centre  of  gravity  from  the  axis,  by  k,  and  we  shall  have, 

2(mr)  =  Mh (  101.) 

Substituting  this  in  Equation  (100),  we  have, 

2(wr5) 

!=-W <102> 

That  is,  the  distance  from  the  axis  of  suspension  to  the 
axis  of  oscillation  is  equal  to  the  moment  of  inertia,  taken 
with  respect  to  the  axis  of  susjiension,  divided  by  the  moment 
of  the  mass,  taken  icith  respect  to  the  same  axis. 

Let  the  axis  of  oscillation  be  taken  as  an  axis  of  suspen- 
sion, and  denote  its  distance  from  the  new  axis  of  oscillation 
by  I'.  The  distances  of  a,  b  .  .  .  p,  k,  from  0,  will  be 
I  —  r,  I  —  r ',  &c,  and  the  distance  G  0  will  be  /  —  h 
From  the  principle  just  enunciated,  we  shall  have, 

_  Z[W(*-r)-] 
'   _     M(l-k) 


PERIODIC    MOTION.  173 

Or,  performing  the  operation  of  squaring  and  reducing, 

_  Z(?nV-  -  2mrl  +  mr*)  _  2(mP)  -  22(mrl)  -f  2(mr*) 
~  M(l  -  k)  "  _  31(1  -k)  ' 

But  I  is  constant,  hence  2(?nln-)  =  2(m)  x  l2  =  3fP, 
also,  22(?nrl)  =  22(mr)  X  I  =  2Mkl\  from  Equation  (102) 
we  have,  2(mr2)  =  3Ikl.  Substituting  these  values  in  the 
preceding  equation,  we  have, 

_  3IP  -  2MM  +  Mkl  _  31(1  -  k)l 
~  31(1  -  k)  -  M(l  -  k)  ' 

or, 

r  =  i (103.) 

Hence,  it  follows  that  the  axes  of  suspension  and  oscilla- 
tion are  convertible  /  that  is,  if  either  be  taken  as  the  axis 
of  suspension,  the  other  to  ill  be  the  axis  of  oscillation,  and 
the  reverse. 

This  property  of  the  compound  pendulum  has  been  em- 
ployed  to  determine  experimentally  the  length  of  the 
seconds  pendulum,  and  the  value  of  the  force  of  gravity  at 
different  places  on  the  surface  of  the  earth. 

A  straight  bar  of  iron  CD,  is  provided  with  two  knife- 
edge  axes,  A  and  B,  of  hardened  steel,  at  right 
angles  to  the   axis  of  the  bar,  and   having   their 
edges  turned  towards  each  other.      These  axes  are 
so  placed  that  their  plane  will  pass  through  the 
axis  of  the  bar.     The  pendulum  thus  constructed  is 
suspended  on  horixontal  plates  of  polished  agate, 
and  allowed  to  vibrate  about  each  axis  in  turn  till, 
by  filing  away  one  of  the  ends  of  the  bar,  the  times 
of  vibration  about  the  two   axes  are  made  equal. 
The  distance  AB  is  then  equal  to  the  length  of  the  „ 
equivalent  simple  pendulum ;  that   is,  of  a  simple 
pendulum  which  will  vibrate  in  the  same  time  as  the  bai 
about  either  axis. 


171  MECHANICS. 

To  employ  the  pendulum  thus  adjusted  to  find  the  length 
of  a  simple  seconds  pendulum  at  any  place,  the  pendulum  is 
carefully  suspended,  and  allowed  to  vibrate  through  a  very 
small  angle  ;  the  number  of  vibrations  is  counted,  and  the 
time  occupied  is  carefully  noted  by  means  of  a  well-regulated 
chronometer.  The  entire  time  divided  by  the  number  of 
vibrations  performed,  gives  the  time  of  a  single  vibration. 
The  distance  between  the  axes  is  carefully  measured  by  an 
accurate  scale  of  equal  parts,  which  gives  the  length  of  the 
corresponding  simple  pendulum.  To  find  the  length  of  the 
simple  seconds  pendulum,  we  then  make  use  of  Proportion 
(97),  substituting  in  it  for  t'  and  V  the  values  just  found,  and 
for  £,  1  second;  the  only  remaining  quantity  in  the  propor- 
tion is  I,  which  may  be  found  by  solving  the  proportion. 
This  value  of  I  is  the  required  length  of  the  simple  seconds 
pendulum  at  the  place  where  the  observation  is  made.  In 
making  the  observations,  a  variety  of  precautions  must  be 
taken,  and  several  corrections  applied,  the  explanation  of 
which  does  not  fall  within  the  scope  of  this  treatise.  It  is 
only  intended  to  point  out  the  general  method  of  proceed- 
ing. By  a  long  series  of  carefully  conducted  experiments, 
it  has  been  found  that  the  length  of  a  simple  seconds  pen- 
dulum in  the  Tower  of  London  is  3.2616  ft.,  or  39.13921  in. 
By  a  similar  course  of  proceeding,  the  length  of  the  seconds 
pendulum  has  been  determined  for  a  great  number  of  places 
on  the  earth's  surface,  at  different  latitudes,  and  from  these 
results  the  corresponding  values  of  the  force  of  gravity  at 
those  points  have  been  determined  according  to  the  following 
principle : 

Fiom  Equation  (95),  which  is,  t  =  *\/~~'>   we  fin0^  by 

solving  with  respect  to  g,  and  making   1=1, 

o  =  *■*. 

From  this  equation  the  value  of  g  may  be  found  at 
different  places,  by  simply  substituting  for  I  the  length  of  the 


PERIODIC    MOTION.  175 

seconds  pendulum  at  ihose  places.  In  this  manner,  the  value 
m  g  is  found  for  a  great  number  of  places  in  different 
latitudes,  and  from  these  values  the  form  of  the  earth's 
surface  may  be  computed. 

It  has  been  ascertained  in  this  manner  that  if  the  force  of 
gravity  at  any  point  on  the  earth's  surface  be  denoted  by  g, 
the  force  of  gravity  at  a  point  whose  latitude  is  45°,  by  g\ 
and  the  latitude  of  the  place  where  the  force  of  gravity  is 
g\  by  /,  we  shall  have, 

g  —  g\\  —  .002695cos2Z). 


PEACmCAL   APPLICATIONS    OF   THE   PENDULUM. 

125.  One  of  the  most  important  of  the  applications  of 
the  pendulum  is  to  regulating  the  n^ption  of  clocks.  A 
clock  consists  of  a  train  of  wheel  work,  the  last  wheel  of  the 
train  connecting  with  the  upper  extremity  of  a  pendulum- 
rod  by  a  piece  of  mechanism  called  an  escapement.  The 
wheelwork  is  maintained  in  motion  by  means  of  a  descending 
Aveight,  or  by  the  elastic  force  of  a  coiled  spring,  and  the 
wheels  are  so  arranged  that  one  tooth  of  the  last  wheel  in 
the  train  escapes  from  the  upper  end  of  the  pendulum-rod 
at  each  vibration  of  the  pendulum,  or  at  each  beat.  The 
number  of  beats  is  registered  and  rendered  visible  on  a 
dial-plate  by  means  of  indices,  called  the  hands  of  the  clock. 

On  account  of  the  expansion  and  contraction  of  the  ma- 
terial of  which  the  pendulum  is  composed,  the  length  of  the 
pendulum  is  liable  to  continual  variation,  which  gives  rise  to 
an  irregularity  in  the  times  of  vibration  of  the  pendulum. 
To  obviate  this  inconvenience,  and  to  render  the  times  of 
vibration  perfectly  uniform,  several  ingenious  devices  have 
been  resorted  to,  giving  rise  to  what  are  called  compensating 
pendulums.  We  shall  indicate  two  of  the  most  important 
of  these  combinations,  observing  that  all  of  the  remaining 
ones  are  nearly  the  same  in  principle,  differing  only  in  the 
modes  of  application. 


176 


MECHANICS. 


Graham's  Mercurial  Pendulum. 

126.  Graham's  mercurial  pendulum  consists  of  a  rod  oi 
steel  about  42  inches  long,  branched  towards  its  lower  end, 
so  as  to  embrace  a  cylindrical  glass  vessel  V  or  8  inches  deep, 
and  having  6.8  in.  of  this  depth  filled  with  mercury.  The 
exact  quantity  of  mercury  being  dependent  on  the  weight 
and  expansibility  of  the  other  parts  oi  the  pendulum,  must 
be  determined  by  experiment  in  each  individual  case 
When  the  temperature  increases,  the  steel  rod  is  lengthened, 
and,  at  the  same  time,  the  mercury  expanding,  rises  in  the 
cylinder.  When  the  temperature  decreases,  the  steel  bar  is 
shortened,  and  the  mercury  falls  in  the  cylinder.  By  a 
proper  adjustment  of  the  quantity  of  mercury,  the  effect 
of  the  lengthening  or  shortening  of  the  rod  is  exactly  coun- 
terbalanced by  the  rising  or  falling  of  the  centre  of  gravity 
of  the  mercury,  arm  the  axis  of  oscillation  is  kept  at  an 
invariable  distance  from  the  axis  of  suspension. 


Harrison's  Gridiron  Pendulum. 

12 "7.  Harrison's  gridiron  pendulum  consists  of  five 
rods  of  steel  and  four  of  brass,  placed  alter- 
nately with  each  other,  the  middle  rod,  or  that 
from  which  the  bob  is  suspended,  being  of  steel. 
These  rods  are  connected  by  cross-pieces  in 
such  a  manner  that,  whilst  the  expansion  of  the 
steel  rods  tends  to  elongate  the  pendulum,  or 
lower  the  bob,  the  expansion  of  the  brass  rods 
tends  to  shorten  the  pendulum,  or  raise  the  bob. 
By  duly  proportioning  the  sizes  and  lengths  of 
the  bars,  the  axis  of  oscillation  may  be  main- 
tained, by  the  combination,  at  an  invariable  dis- 
tance from  the  axis  of  suspension.  From  what 
has  preceded,  it  follows  that  whenever  the  dis- 
tance from  the  axis  of*  oscillation  to  the  axis  of  suspension 
remains  invariable,  the  limes  of  vibration  must  be  abso- 
lutely equal  at  the  same  place.     The  pendulums  just  de- 


t 

Fig.  107. 


PERIODIC   MOTION.  177 

scribed  are  principally  used  for  astronomical  clocks,  where 
great  accuracy  and  great  uniformity  in  the  measure  of  time 
is  indispensable. 

Basis  of  a  system  of  Weights  and  Measures. 

128.  The  pendulum  is  of  further  importance,  in  a  prac- 
tical point  of  view,  in  furnishing  the  standard  of  comparison 
which  has  been  made  use  of  as  a  basis  of  the  English  system 
of  weights  and  measures.  The  length  of  the  seconds  pendu- 
lum at  any  place,  can  always  be  found,  and  it  must  always 
be  the  same  at  that  place.  We  have  seen  that  this  length 
was  determined,  with  great  accuracy,  in  the  Tower  of  Lon- 
don, to  be  3.2616  ft.  It  has  been  decreed  by  the  British 
Government,  that  the  3,27n  etn  Part  °f  tne  lengtn  °f  tne 
simple  seconds  pendulum,  in  the  Tower  of  London,  shall  be 
regarded  as  a  standard  foot.  From  this,  by  multiplication 
and  division,  every  other  unit  of  lineal  measure  may  be  de- 
rived. By  constructing  squares  and  cubes  upon  the  linear 
units,  we  at  once  arrive  at  the  units  of  area  and  of  volume. 

It  has  further  been  decreed,  that  a  cubic  foot  of  distilled 
water,  at  the  temperature  of  maximum  density,  shall  be  re- 
garded as  weighing  1000  standard  ounces.  This  fixes  the 
ounce  ;  and  by  multiplication  and  division,  all  other  units  of 
weight  may  be  derived. 

This  system  enables  us  to  refer  to  the  original  standard, 
when,  from  any  circumstances,  doubt  may  exist  as  to  the 
accuracy  of  standard  measures.  Even  should  every  vestige 
of  a  standard  be  swept  from  existence,  they  might  be  per- 
fectly restored,  by  the  process  above  indicated. 

The  American  system  of  weights  and  measures  is  adopted 
from  that  of  Great  Britain,  and  is,  in  all  respects,  the  same 
as  that  above  described. 

EXAMPLES. 

1.  The  length  of  a  seconds  pendulum  is  39.13921  hi.  If 
it  be  shortened  0.130464  in.,  how  many  vibrations  will  be 
gained  in  a  day  of  24  hours  ? 

8* 


ITS  MECHANICS. 

SOLUTION. 

The  times  of  vibration  of  two  pendulums  at  the  same 
place,  are  to  each  other  as  the  square  roots  of  their  lengths 
(Eq.  97).  Hence,  the  number  of  vibrations  made  in  any 
given  time,  are  inversely  proportional  to  the  square  roots  of 
their  lengths.  If,  therefore,  Ave  denote  the  number  of  vi- 
brations gained  in  24  hours,  or  86400  seconds,  by  a-,  we 
shall  have, 

86400  :  86400  -f  x  :  :   ^39-008747  :    v/39.13921 ; 
or,    86400  :  86400  -f  x  :  :  6.2457  :  6.2561. 

Whence,  x  —  144,  nearly.  Ans. 

2.  A  seconds  pendulum  being  carried  to  the  top  of  a 
mountain,  was  observed  to  lose  5  vibrations  per  day  of 
86400  seconds.  Required  the  height  of  the  mountain, 
reckoning  the  radius  of  the  earth  at  4000  miles. 

SOLUTION. 

The  squares  of  the  times  of  vibration,  at  any  two  points, 
are  inversely  proportional  to  the  forces  of  gravity  at 
those  points  (Eq.  98).  But  the  forces  of  gravity  at  the 
same  points  are  inversely  as  the  squares  of  their  distances 
from  the  centre  of  the  earth.  Hence,  the  times  of  vibration 
are  proportional  to  the  distances  of  the  points  from  the  cen- 
tre of  the  earth;  and,  consequently,  the  number  of  vibra- 
tions in  any  given  time,  as  24  hours,  for  example,  will  be 
inversely  as  those  distances.  If,  therefore,  we  denote  the 
height  of  the  mountain  in  miles  by  ic,  we  shall  have, 

86400  :  86405  :  :  4000  :  4000  +  x. 

Whence,  x  =  f£$$!  =  0.2315  miles,  or,  1222  feet.  Ans. 

3.  What  is  the  time  of  vibration  of  a  pendulum  whose 
length  is  GO  inches,  when  the  force  of  gravity  is  reckoned  at 
32£  ft?  Ans.    1.2387  sec. 


PERIODIC    MOTION.  179 

4.  How  many  vibrations  will  a  pendulum  36  inches  in 
length  make  in  one  minute,  the  force  of  gravity  being  the 
same  as  before  ?  A?is.  62.53. 

5.  A  pendulum  is  found  to  make  43170  vibrations  in  12 
hours.  How  much  must  it  be  shortened  that  it  may  beat 
seconds  ? 

SOLUTION. 

We  shall  have,  as  in  Example  1st, 


43170  :  43200  :  :   -v/39.13921   :   -y/39-13921  +  x- 
Whence,  x  =  0.0544  in.     Ans. 

6.  In  a  certain  latitude,  the  length  of  a  pendulum  vi- 
brating seconds  is  39  inches.  What  is  the  length  of  a  pen- 
dulum vibrating  seconds,  in  the  same  latitude,  at  the  height 
of  21000  feet  above  the  first  station,  the  radius  of  the  earth 
being  3960  miles?  Ans.    38.9218  in. 

7.  If  a  pendulum  make  40000  vibrations  in  6  hours,  at 
the  level  of  the  sea,  how  many  vibrations  will  it  make  in  the 
same  time,  at  an  elevation  of  10560  feet  above  the  same 
point,  the  radius  of  the  earth  being  3960  miles? 

Ans.    39979.8. 
Centre  of  Percussion. 

129.     The  point  O,  Fig.  108,  is  a  point  at  which,  if  the 
entire  mass  were  concentrated,  and  the  im- 
pressed forces  applied  to  it,  the  effect  produced 
would  be  in  nowise  different  from  what  actu- 
ally obtains.     Were  an  impulse  applied  at  this  , 
point,  capable  of  generat  ng  a  quantity  of  mo-            /' 
tion  equal  and  directly  opposed  to  the  resul-          /-.. 
tant  of  all   the  quantities  of  motion  of  the        '  """ 
particles  of  the  body,  at  any  instant,  the  body         Fig.  ios. 
would  evidently  be  brought  to  a  state  of  rest 
without   imparting  any  shock   to  the   axis   of  suspension. 
The  direction  of  the  impulse  remaining  the  same  as  before, 


180  MECHANICS. 

no  matter  what  may  be  its  intensity,  there  will  stiL  be  no 
shock  on  the  axis.  This  point  is,  therefore,  called  the  centre 
of  percussion.  We  may  then  define  the  centre  of  percus- 
sion to  be  that  point  of  a  body  restrained  by  an  axis,  at 
which,  if  the  body  be  struck  in  a  direction  perpendicular  to 
a  plane  passed  through  this  point  and  the  axis  of  suspension, 
n  >  -hock  will  be  imparted  to  the  axis.  It  is  a  matter  of 
common  observation  that,  if  a  rod  held  in  the  hand 
be  struck  at  a  certain  point,  the  hand  will  not  feel  the 
blow,  but  if  it  be  struck  at  any  other  point  of  its  length, 
there  will  be  a  shock  felt,  the  intensity  of  which  will  depend 
upon  the  intensity  of  the  blow,  and  upon  the  distance  of  its 
point  of  application  from  the  first  point. 

Moment  of  Inertia. 
130.  The  moment  of  inertia  of  a  body  with  respect  to 
an  axis,  is  the  ahjebraic  sum  of  the  products  obtained  by 
multiplying  the  mass  of  each  elementary  particle  by  the 
square  of  its  distance  from  the  axis.  Denoting  the  moment 
of  inertia  with  respect  to  any  axis,  by  A",  the  mass  of  any 
element  of  the  body,  by  m,  and  its  distance  from  the  axis, 
by  r,  we  have,  from  the  definition, 

K  =  Z(mr)      ....      (104.) 

The  moment  of  inertia  evidently  varies,  in  the  same  body, 
according  to  the  position  of  the  axis.     To  investigate  the 
law  of  variation,  let  AB  represent  any  sec- 
tion of  the  body  by  a  plane  perpendicular 
to  the  axis;  C,  the  point  in  which  this  plane 
cuts  the  axis;  and  G,  the  point  in  which  it 
cuts  a  parallel   axis  through  the  centre  of 
gravity.     Let    P   be   any   element    of  the 
body,  whose  mass  is  ?/?,  and  denote  7'(7by  i 
CG  by  k. 

From   the  triangle    CPG,   accordin 
Trigonometry,  we  have, 

r-  =  *a  +  A;3  —  2skcosCGP 


1  ^ 

B 

Fig.  109. 

PG  by  v,  and 

i  principle  c 

f 

MOMENT    OF    INERTIA.  181 

Substituting  in  (104),  and  separating  the  terms,  we  have, 

K  —  2(msn-)  +  Kmtf)  —  2I(mskcosCGP). 

Or,  since  k  is  constant,  and  l(m)  =  Mz  the  mass  of  the 
entire  body,  we  have, 

K '=  2{ms*)  +  Mlt  -  2kl(mscosCGP). 

But  scosCGP  =  GIT,  the  lever  arm  of  the  mass  -/i, 
with  respect  to  the  axis  through  the  centre  of  gravity. 
Hence,  2(ms  cosCGP),  is  the  algebraic  sum  of  the  mo- 
ments of  all  the  particles  of  the  body  with  respect  to  the 
axis  through  the  centre  of  gravity ;  but  from  the  principle 
of  moments,  this  is  equal  to  0.     Hence, 

K=  2(msa)  +  Mk*   .     .     .     (105.) 

The  first  term  of  the  second  member  of  (105),  is  the  ex- 
pression for  the  moment  of  inertia,  with  respect  to  the  axis 
through  the  centre  of  gravity. 

Hence,  the  moment  of  inertia  of  a  body  with  respect  to 
any  axis,  is  equal  to  the  moment  of  inertia  tcith  respect  to 
a  parallel  axis  through  the  centre  of  gravity,  plus  the  mass 
of  the  body  into  the  square  of  the  distance  between  the  two 
axes. 

The  moment  of  inertia  is,  therefore,  the  least  possible, 
when  the  axis  passes  through  the  centre  of  gravity.  If  any 
number  of  parallel  axes  be  taken  at  equal  distances  from  the 
centre  of  gravity,  the  moment  of  inertia  with  respect  to 
each,  will  be  the  same. 

The  moment  of  inertia  of  a  body  with  respect  to  any 
axis,  may  be  determined  experimentally  as  folloAvs.  Make 
the  axis  horizontal,  and  allow  the  body  to  vibrate  about  it, 
as  a  compound  pendulum.  Find  the  time  of  a  single  vibra- 
tion, and  denote  it  by  t.  This  value  of  t,  in  Equation  (95), 
makes  known  the  value  of  I.  Determine  the  centre  of 
gravity  by  some  one  of  the  methods  given,  and  denote  its 


182  MECHANICS. 

distance  from  the  axis,  by  k.     Find  the  mass  of  the  body 
(Art.  11),  and  denote  it  by  M. 
We  have,  from  Equation  (102), 

MM  =  Z(mr-)  =  K. 

Substituting  for  M,  Z,  and  A-,  the  values  already  found, 
and  the  value  of  A"  will  be  the  moment  of  inertia,  with  res- 
pect to  the  assumed  axis.  Subtract  from  this  the  value  of 
Jfk',  and  the  remainder  will  be  the  moment  of  inertia 
with  respect  to  a  parallel  axis  through  the  centre  of  gravity. 
The  moment  of  inertia  of  a  homogeneous  body  of  regular 
figure,  is  most  readily  found  by  means  of  the  calculus  A 
few  examples  of  the  application  of  the  calculus  to  finding 
the  moment  of  inertia  of  bodies  are  subjoined. 

Application  of  the  Calculus  to  determine  the  Moment  of  Inertia. 

131.  To  render  Formula  (104)  suitable  to  the  application 
of  the  calculus,  we  have  simply  to  change  the  sign  of  sum- 
mation, 2,  to  that  of  integration,  /,  and  to  replace  in  by 
dM,  and  r  by  x.     This  gives, 

E  =  ftfdM (106.) 

Example  1.  To  find  the  moment  of  inertia  of  a  rod  or  bar 
of  uniform  thickness  with  respect  to  an  axis  through  its 
centre  of  gravity  and  perpendicular  to  the  length  of  the 
rod. 

Let  AB  represent  the  rod,  G  its  centre  of  gravity,  and 
E    any    element    contained    by 

planes    at    right    angles   to    the         [2 jp 

length  of  the  rod  and  infinitely       A  ,      M 

near    each    other.      Denote    the 

mass  of  the  rod  by  3/,  its  length, 

by  27,  the  distance  GE,  by  #,  and  R    110 

the  thickness  of  the  element  A7, 

by  dx.     Then  will  the  mass  of  the  element  E  be  equal  to 


MOMENT    OF    INERTIA. 


183 


M 

21 


dx.     Substituting  this  for  d3f,  in  Equation  (106),  and 


integrating  between  the  limits  —  I  and    -f  I,  we  have, 


+  i 


*M 


K  =   /  -r  x'dx  =  31^- 
J   21 


—  i 


For  any  parallel  axis  whose  distance  from  G  is  (7,  we  shall 
have, 


r(|  +  tf 


(107.) 


These  two  formulas  are  entirely  independent  of  the 
breadth  of  the  filament  in  the  direction  of  the  axis  DC. 
They  will,  therefore,  hold  good  when  the  filament  AB  is 
replaced  by  the  rectangle  KF.  In  this  case,  M  becomes 
the  mass  of  the  rectangle,  21  the  length  of  the  rectangle, 
and  d  the  distance  of  the  centre  of  gravity  of  the  rectangle 
from  the  axis  parallel  to  one  of  its  ends. 

Example  2.  To  find  the  moment  of  inertia  of  a  thin 
circular  plate  about  one  of  its  diameters. 

Let  A  CB  represent  the  plate,  AB  the  axis,  and   CD' 

any  element  parallel  to  AB.     Denote 

the  radius  0(7,  by  r,  the  distance  OE, 

by  jb,  the  breadth  of  the  element  EF, 

by  dx,  and  its  length  D  (7,  by  2y.     If 

Ave    denote   the    entire   mass   of   the 

plate,  by  M,  the  mass  of  the  element 

2xi  dx 
CD   will  be   equal  to  JLT-~-;    or, 

we     have, 


since 


y   —    yr    —  ar 


dM  =  M 


2yfl 


*r' 


dx. 


Substituting  in  Equation  (106),  we  have, 


184 


MECHANICS. 


M 


,J 


K=f--l.x>(r>-x>) 


dx. 


Integrating  by  the  aid  of  Formulas  A  and  JB  (Integral 
Calculus),  and  taking  the  integral  between  the  limits 
x  =  —  r,    and   x  =  +  r,    we  find, 


K  =  M 


4  ' 


and  for  a  parallel  axis  at  a  distance  from  AB  equal  to  d, 
JT'  =  Jf(^  +  ^) (108.) 

Example  3.  To  find  the  moment  of  inertia  of  a  circular 
plate  with  respect  to  an  axis  through 
its  centre  perpendicular  to  the  face  of 
the  plate. 

Let  the  dimensions  and  mass  of  the 
plate  be  the  same  as  before.  Let  KL 
be  an  elemetary  ring  whose  radius  is  x, 
and  whose  breadth  dx.  Then  will  the 
mass  of  the  elementary  ring  be  equal 


to  M 


2txdx 


or    dM  — 


iMxdx 


Substituting     this    in    Equation    (106),     and    taking    the 
integral  between  the  limits    x  —  0,    and   x  =  r,    we  have, 


^  r23fxsdx       Mr' 

o 


For  a  parallel  axis  at  a  distance  d  from  the  primitive 
axis, 


JT' =  Jf  (y  +  <*») (109.) 


MOMENT    OF    INERTIA. 


185 


Example  4.  To  find  the  moment  of  inertia  of  a  circular 
ring,  such  as  maybe  generated  by  revolving  a  rectangle  about 
a  line  parallel  to  one  of  its  sides, 
taken  with  respect  to  an  axis  through 
the  centre  of  gravity  and  perpendi- 
cular to  the  face  of  the  ring.  This  case 
differs  but  little  from  the  preceding. 
Denote  the  inner  radius  by  r,  the 
outer  radius  by  r\  and  the  mass  of 
the  ring  by  31.  If  we  take,  as  before, 
an  elementary  ring  whose  radius  is 
sp,  and  whose  breadth  is  dx,  we  shall  have  for  its  mass, 


Fig.  113. 


dM  =  M 


2xdx 


Substituting   in  Equation   (106),  and   integrating  between 
the  limits  r,  and  r\  we  have, 


A  =  fM^—p  =  M-^—)  =  M~— 


For  a  parallel  axis  at  a  distance  from  the  primitive  axis 
equal  to  c?,  we  have. 


r'2  +  r1 


->  =  M(^L+(r) 


(no..; 


If  in  these  values  of  K  and  K'  we  make  r  =  0,  we  shall 
deduce  the  results  of  the  last  example. 


Example  5.  To  find  the  moment  of  inertia  of  a  right 
cylinder  with  respect  to  an  axis  through  the  centre  of 
gravity  and  perpendicular  to  the  axis  of  the  cylinder. 

Let  AB  represent  the  cylinder,    CD   the  axis  through 


c 

E 


iSo*  MECHANIC8. 

its  centre  of  gravity,  and  E  an  ele- 
ment of  the  cylinder  between  two 
planes  perpendicular  to  the  axis,  and 
distant  from  each  other,  by  dx.  De- 
note the  length  of  the  cylinder  by  2/, 
the   area  of  its  cross  section  by  ^r2,  Ficr  U4 

r  being  the  radius  of  the   base;  the 

distance  of  the  section  E  from  the  centre  of  gravity,  by  x, 
and  the  mass  of  the  cylinder,  by  M. 

dx 
The  mass   of   the    element    E   is  equal  to  M-*      Its 

moment  of  inertia  with  respect  to  its  diameter  parallel  to 

^t^  •  ^        Mdx      r3   ,_  ,  _ 

CD,  is  equal  to  — —  x  —  (Example  2),  and  with   respect 

to  CD  parallel  to  it,  — —  I h  x M  • 

Integrating  this  expression  between  the  limits  x  =  —  /, 
and  x  —  -f-  I,  we  have, 


*=7*(f +"■)*=<+ J) 


For   an   axis  parallel    to   the  primitive    one,    and    at  a 
distance  from  it  equal  to  d, 


Centre  of  Gyration. 

132.  The  centre  of  gyration  of  a  body  with  respect  to 
an  axis,  is  a  point  at  which,  if  the  entire  mass  be  concen- 
trated, its  moment  of  inertia  will  remain  unchanged.  The 
distance  from  this  point  to  the  axis  is  called  the  radius  of 
gyration* 


MOMENT    OF    INERTIA.  187 

Let  M  denote  the  mass  of  the  body,  and  k'  its  radius  of 
gyration ;  then  will  the  moment  of  inertia  of  the  concen 
tinted  mass  with  respect  to  the  axis, be  equal  to  Mk'2 ;  but 
this  must,  by  definition,  be  equal  to  the  moment  of  inertia 
with  respect  to  the  same  axis,  or  2(wir2) ;  hence, 

That  is,  the  radius  of  gyration  is  equal  to  the  square 
root  of  the  quotient  obtained  by  dividing  the  moment  of 
inertia  with  respect  to  the  same  axis,  by  the  entire  ?nass. 

Since  M  is  constant  for  the  same  body,  it  follows  that  the 
radius  of  gyration  will  be  the  least  possible  when  the 
moment  of  inertia  is  the  least  possible,  that  is,  when  the 
axis  passes  through  the  centre  of  gravity.  This  minimum 
radius  is  called  the  principal  radius  of  gyration.  If  we 
denote  the  principal  radius  of  gyration  by  &,  we  shall  have, 
from  the  examples  of  Article  (131),  the  following  results: 


Example  1,     .     k'  =  v/-r  +  dr  ;  k  =  I  y^J 


/r2  r 

Example  2,     .     k'  =  \I  —  -f  d2 ;  k  =  - 


Example  3,     .     k'  =  \J  —  -j-  d* ;  k  =  r-y/J. 


/r'2  -f-  r2  /r'2  4-  r2 
+  d2 ;      k  =  \f  — — — 


Example^     .     ^  =  y/£  +  |  +  #;       k  =  yjj  + 


188 


MECHANICS. 


CHAPTER  VI. 


CURVILINEAR   AND    ROTARY    MOTION. 


Motion  of  Projectiles. 

133.  If  a  body  is  projected  obliquely  upwards  in 
vacuum,  and  then  abandoned  to  the  force  of  gravity,  it  will 
be  continually  deflected  from  a  rectilinear  path,  and,  after 
describing  a  curvilinear  trajectory,  will  finally  reach  the 
horizontal  plane  from  which  it  started. 

The  starting  point  is  called  the  point  of  projection ;  the 
distance  from  the  point  of  projection  to  the  point  at  which 
the  projectile  again  reaches  the  horizontal  plane,  through 
the  point  of  projection,  is  called  the  range,  and  the  time 
occupied  is  called  the  time  of  flight.  The  only  forces  to  be 
considered,  are  the  initial  im- 
pulse and  the  force  of  gravity. 
Hence,  the  trajectory  will  lie  in 
a  vertical  plane  passing  through 
the  line  of  direction  of  the 
initial  impulse.  Let  CAB  rep- 
resent this  plane,  A  the  point 
of  projection,  AB  the  range, 
and  AC  a  vertical  line  through 

A.  Take  AB  and  AC  as  co-ordinate  axes;  denote  the 
angle  of  projection  DAB,  by  a,  and  the  velocity  due  to  the 
initial  impulse,  by  v.  Resolve  the  velocity  v  into  two  com- 
ponents, one  in  the  direction  A  C,  and  the  other  in  the 
direction  AB.  We  shall  have,  for  the  former,  rsina,  and, 
for  the  latter,  v  cost. 

The  velocities,  and,  consequently,  the  spaces  described  in 
the  direction  of  the  co-ordinate  axes,  will  (Art.  18)  be  en- 
tirely   independent    of   each    other.      Denote    the    space 


Fig.  115. 


OURVILINEAR    AND    ROTARY    MOTION.  189 

described  in  the  direction  A  (7,  in  any  arbitrary  time  £,  by  y. 
The  circumstances  of  motion  in  this  direction,  are  those  of  a 
body  projected  vertically  upwards  with  an  initial  velocity 
v  sina,  and  then  continually  acted  upon  by  the  force  of 
gravity.  Hence,  Equation  (78)  is  applicable.  Making,  in 
that  equation,  h  =  y,    and   v'  =  v  sina,  we  have, 

y  =  vsmat  —  \gf  ,     .     .     .     (113.) 

Denote  the  space  described  in  the  direction  of  the  axis 
AB,  in  any  arbitrary  time  t,  by  x.  The  only  force  acting 
in  the  direction  of  this  axis,  is  the  component  of  the  initial 
impulse.  Hence,  the  motion  in  the  direction  ot  the  axis  of 
x  will  be  uniform,  and  Equation  (55)  is  applicable.  Making 
s  =  x,     and    v  =  v  cosa,   we  have, 

x  =  v  cosa  t (114>) 

If  we  suppose  t  to  be  the  same  in  Equations  (113)  and 
(114),  they  will  be  simultaneous,  and,  taken  together,  will 
make  known  the  position  of  the  projectile  at  any  instant. 

From  (114),  we  have, 


t  - 


V  COSa 


which,  substituted  in  (113),  gives, 


sina  qx*  , 

y  =  — *-  5-r—r  •    •    •    (115-) 

cosa  2u2COS  a  x 


an  equation  which  is  entirely  independent  of  t.  It,  there- 
fore, expresses  the  relation  between  x  and  y  for  any  value 
of  t  whatever,  and  is,  consequently,  the  equation  of  the  tra- 
jectory. Equation  (115)  is  the  equation  of  a  parabola 
whose  axis  is  vertical.  Hence,  the  required  trajectory  is  a 
parabola. 


190  MECHANICS. 

To  find  an  expression  for  the  range,  make  y  =  0,  in  (115), 
and  deduce  the  corresponding  value  of  x.  Placing  the  value 
of  y  equal  to  0,  we  have, 

sina  ax* 

cosa  2y2cos-a 

2y2sina  cosa 
.*.     x  =  0,      and     x  = • 


The  first  value  of  x  corresponds  to  the  point  of  projection, 
and  the  second  is  the  value  of  the  range,  AB. 
From  trigonometry,  we  have, 

2sina  cosa  =  sin2a. 

If  we  denote  the  height  due  to  the  initial  velocity,  by  A, 
we  shall  have, 

v*  —  2gh. 

Substituting  these  in  the  second  value  of  ic,  and  denoting 
the  range  by  r,  we  have, 

r  =  2Asin2a (  116.) 

The  greatest  value  of  r  will  correspond  to  the  value 
a  =  45°,  in  which  case,  2a  —  90°,  and  sin  2a  =  1. 
Hence,  we  have,  for  the  greatest  range, 

r  =  2h.  # 

That  is,  it  is  equal  to  twice  the  height  due  to  the  initio:* 
velocity. 

If,  in  (116),  we  replace  a  by  90°  —  a,  we  shall  have, 

r  —  2Asin(180°  —2a)  =  2A  sin2a, 

the  same  value  as  before.     Hence,  we  conclude  that  there 
are  two  angles  of  projection,  complements  of  each  other, 


CURVILINEAR    AND    ROTARY    MOTION. 


191 


which  give  the  same  range.  The  trajectories  in  the  two 
cases  are  not  the  same,  as  may  be  shown  by  substituting  the 
values  of  a,  and  90°  —  «,  in  Equation  (115).  The  greater 
angle  of  projection  gives  a  higher  elevation,  and,  conse- 
quently, the  projectile  descends  more  vertically.  It  is  for 
this  reason  that  the  gunner  selects  the  greater  of  the  two 
angles  of  elevation  when  he  desires  to  crush  an  object, 
and  the  lesser  one  when  he  desires  to  batter,  or  overturn 
the  object.  If  a  =  90°,  the  value  of  r  becomes  0.  That 
is,  if  a  body  be  projected  vertically  upwards,  it  will  return 
to  the  point  of  projection. 

To  find  the  time  of  flight,  make  x  =  r,  in  Equation  (114), 
and  deduce  the  corresponding  value  of  t.     This  gives, 


t  = 


V  COS* 


(117.) 


The  range  being  the  same,  the  time  of  flight  will  be 
greatest  when  a  is  greatest.  Equation  (114)  also  gives  the 
time  required  for  the  body  to  describe  any  distance  in  the 
direction  of  the  horizontal  line  AB. 

In  Equation  (117)  there  are  four  quantities,  t,  r,  zyand  a, 
and  from  it,  if  any  three  are  given,  the  remaining  one  may 
be  determined. 

As  an  application  of  the  principles  just  deduced,  let  it  be 
required  to  determine  the  angle 
of  projection,  in  order  that  the 
projectile  may  strike  a  point 
II,  at  a  horizontal  distance 
AG  =  x'  from  the  point  of 
projection,  and  at  a  height 
Gil  =  y'  above  it. 

Since  the  point  H  lies  on  the  Fig.  lie. 

trajectory,  its  co-ordinates  must 
satisfy  the  equation  of  the  curve,  giving 


y'  —  a;' tana 


9* 


2v'cos,a 


192  MECHANICS. 

From  trigonometry,  we  have, 

1  1 


cos  a  = 


sec2  a         1  4-  tan2  a 


Substituting    this  in    the  preceding   equation,  we  have* 
after  clearing  of  fractions, 

2v2y'  =  2uVtana  —  gxn{\  +  tai\2a) ; 

or,  transposing  and  reducing, 


2u2  2i>V  +  gx'2 

tan2a  —  — -  tana  — — 

gx  gx* 


Hence, 


v*  /  v*         2»V 

tana  =  — r  ±  \/  ~rT* 

gx'       V  g'x 3  g 


+  ax» 


gx'  -"  V  </V2  gx'*        ■ 

or,  making  v*  =  2gh, 

2h  /V?       ±hy'+x'a       2h  ±  ^/W-4hg  — ai7* 

tana  —  —r±  \/  —jx ■ — ,i = "77 " 

x        V   xr  x  x' 

This  shows  that  there  are,  in  general,  two  angles  of  pro- 
jection, under  either  of  which  the  point  may  be  struct. 
If  we  suppose 

xn  =  4A2  -  4hg'   ....     (118.) 

the  quantity  under  the  radical  sign  will  be  0,  and  the  two 
angles  of  projection  will  become  one. 

But  if  x'  and  y'  be  regarded  as  variables,  Equation  (118) 
represents  a  parabola  whose  axis  is  a  vertical  passing 
through  the  point  of  projection.  Its  vertex  is  at  a  distance 
above  the  point  A,  equal  to  A,  its  focus  is  at  A,  and  its 
parameter  is  equal  to  4/<,  or  twice  the  range. 

If  we  suppose 

x"  <  ih2    -  4hg\ 


CURVILINEAR    AND    ROTARY    MOTION. 


193 


the  point  (&',  y'),  will  lie  within  the  parabola  just  described, 
the  quantity  under  the  radical  sign  will  be  positive,  and 
there  will  be  two  real  values  of  tan  a,  and,  consequently, 
two  angles  of  projection,  under  either  of  which  the  point 
may  be  struck. 
If  we  suppose 

xn  >  4A2  -  4Ay', 

the  point  (#',  y'),  wtII  be  without  this  parabola,  the  values 
of  tana  will  both  be  imaginary,  and  there  will  be  no  angle 
under  which  the  point  can  be  struck. 


B'  21l  A.         2)i 

Fig.  117. 

Let  the  parabola  B'LB  represent  the  curve  whose  equa- 
tion is 

sb"  =  4A2  -  ±hy'. 

Conceive  it  to  be  revolved  about  AL,  as  an  axis  generat- 
ing a  paraboloid  of  revolution.  Then,  from  what  has  preced- 
ed, we  conclude,  first,  that  every  point  lying  within  the 
surface  may  be  reached  from  A,  with  a  given  initial  velocity, 
under  two  different  angles  of  projection  ;  second,  that  every 
point  lying  on  the  surface  can  be  reached,  but  only  by  a  sin- 
gle angle  of  projection  ;  thirdly,  that  no  point  lying  without 
the  surface  can  be  reached  at  all. 

If  we  suppose  a  body  to  be  projected  horizontally  from  an 
elevated  point  A,  the  trajectory  will  be 
made  known  by  Equation  (115)  by  sim- 
ply making  a  =  0  ;  whence,  sina  =  0, 
and  cos*  =  1.  Substituting  and  reduc- 
ing, we  have, 

y  =  -  g-- 

9 


(119.) 


Fig.   118 


194  MECHANICS. 

For  every  value  of  #,  y  is  negative,  which  shows  that 
every  point  of  the  trajectory  lies  below  the  horizontal  line 
through  the  point  of  projection.  If  we  suppose  ordinates  to 
be  estimated  positively  downwards,  we  shall  have, 


y=g      ....     (120.) 


To  find  the  point  at  which  the  trajectory  will  reach  any 
horizontal  plane  B  C,  whose  distance  below  the  point  A  is 
A',  we  make  y  =  h\  in- (120),  whence, 

x  =  BC  =  vJ—     .     .     .     (121.) 

V    g 

On  account  of  the  resistance  of  the  air,  the  results  of  the 
preceding  discussion  will  be  greatly  modified.  They  will, 
however,  approach  more  nearly  to  the  observed  phenomena, 
as  the  velocity  is  diminished  and  the  density  of  the  projec- 
tile increased.  The  atmospheric  resistance  increases  as  the 
square  of  the  velocity,  and  as  the  cross  section  of  the  pro- 
jectile exposed  to  the  action  of  the  resistance.  In  the  air, 
it  is  found  that,  under  ordinary  circumstances,  the  maximum 
range  is  obtained  by  an  angle  of  projection  not  for  from 
34°. 

E  X  A  MPLE  S. 

1.  "What  is  the  time  of  flight  of  a  projectile,  when  the 
angle  of  projection  is  45°,  and  the  range  6000  feet? 

SOLUTION. 

"When  the  angle  of  projection  is  45°,  the  range  is  equal  to 
twice  the  height  due  to  the  velocity  of  projection.  Denot- 
ing this  velocity  by  tf,  we  shall  have, 

V7  =  2gh  =  2   X  321   x  3000  =  193000. 


CURVILINEAR    AND    ROTARY    MOTION.  195 

Whence,  we  find, 

v  =  439.3  ft. 

From  Equation  (117),  we  have, 


r  6000  ,rt„ 

t  — —  =  ■— — — -  =  19.3  sec.  Ans, 

vcosa         439.3  cos4o° 


2.  What  is  the  range  of  a  projectile,  when  the  angle  of 
projection  is  30°,  and  the  initial  velocity  200  feet  ? 

Ans.  1076.9  ft. 

3.  The  angle  of  projection  under  which  a  shell  is  thrown 
is  32°,  and  the  range  3250  feet.     What  is  the  time  of  flight  ? 

Ans.  11.25  sec,  nearly. 

4.  Find  the  angle  of  projection  and  velocity  of  projec- 
tion of  a  shell,  so  that  its  trajectory  shall  pass  through  two 
points,  the  co-ordinates  of  the  first  being  x  =  1700  ft., 
y  =  10  ft.,   and  of  the  second,   x  =  1800  ft.,   y  =  10  ft. 

SOLUTION. 

Substituting  for  x  and  y,  in  Equation  (115),  (1700,10), 
and  (1800, 10),  we  have, 

10  =  1700tan«-<«; 

2y2cos2a 

and, 

10  =  ISOOtan*   -   £» 
2trcosa 


Finding  the  value  of  — ^ — —    from  each  of  these  equa- 
&  2v2COS2a  ^ 

tions,  and  placing  the  two  equal  to  each  other,  we  have, 

after  reduction, 

(18)2(l-170tana)  =  (I7)2(l-180tana). 


196  MECHANICS. 

Whence,  by  solution, 
tana  =  -g^  =  0.01144,  nearly  ;         .'.     a  =  39'  19". 

We  have,  from  trigonometry, 

1  1  374544  *     . 

cos'a  = = —    =  =  .99987. 

sec'a  l+tan2a  374593 

Substituting  for  tana  and  cosa  in  the  first  equation 
their  values  as  just  deduced,  we  find,  for  ir2, 

«,  = (1V00)V  =   92961666  = 

2cos3a(1700tana— 10)  18.89 

Whence, 

v  =  2218.3  ft. 

The  required  angle  of  projection  is,  therefore,  39'19",  and 
the  required  initial  velocity,  2218.3  ft. 

4.  At  what  elevation  must  a  shell  be  projected  with  a 
velocity  of  400  feet,  that  it  may  range  7500  feet  on  a  plane 
which  descends  at  an  angle  of  30  ? 

SOLUTION. 

The  co-ordinates  of  the  point  at  which  the  shell  strikes,  are 

x'  —  7500cos30°  =  6495  ;  and  y'  —  —  7500sin30°  =  —  3750. 

And  denoting  the  height  due  to  the  velocity  400  ft.,  by  h, 

we  have, 

v1 
h  -   —  =  2486  ft. 
2<7 

Substituting  these  values  in  the  formula, 


2A  ±   J\h%  —  ihy*  —  x" 
tana  ~ ; i , 


CURVILINEAR    AND    ROTARY    MOTION.  197 

and  reducing,  we  have, 

4972  ±  4453 

tana  =  

6495 

Hence,    a  =  4°  34'  10",     and   55°  25'  41".   Ans. 

Centripetal  and  Centrifugal  Forces. 

134.  Curvilinear  motion  can  only  result  from  the  action 
of  an  incessant  force,  whose  direction  differs  from  that  of 
the  original  impulse.  This  force  is  called  the  deflecting 
force,  and  may  arise  from  one  or  more  active  forces,  or  it 
may  result  from  the  resistance  offered  by  a  rigid  body,  as 
when  a  ball  is  compelled  to  run  in  a  curved  gr©ove.  What- 
ever may  be  the  nature  of  the  deflecting  forces,  we  can 
always  conceive  them  to  be  replaced  by  a  single  incessant 
force  acting  transversely  to  the  path  of  the  body.  Let  the 
deflecting  force  be  resolved  into  two  components,  one  nor- 
mal to  the  path  of  the  body,  and  the  other  tangential  to  it. 
The  latter  force  will  act  to  accelerate  or  retard  the  motion 
of  the  body,  according  to  the  direction  of  the  deflecting 
force ;  the  former  alone  is  effective  in  changing  the  direction 
of  the  motion.  The  normal  component  is  always  directed 
towards  the  concave  side  of  the  curve,  and  is  called  the 
centripetal  force.  The  body  resists  this  force,  by  virtue  of 
its  inertia,  and,  from  the  law  of  inertia,  the  resistance  must 
be  equal  and  directly  opposed  to  the  centripetal  force.  This 
force  of  resistance  is  called  the  centrifugal  force.  Hence, 
we  may  define  the  centrifugal  force  to  be  the  resistance 
which  a  body  offers  to  a  force  which  tends  to  deflect  it  from 
a  rectilineal  path.  The  centripetal  and  centrifugal  forces 
taken  together,  are  called  central  forces. 

Measure  of  the  Centrifugal  Force. 

135.  To  deduce  an  expression  for  the  measure  of  the 
centrifugal  force,  let  us  first  consider  the  case  of  a  single 
material  point,  which  is  constrained  to  move  in  a  circular 


198 


MECHANICS. 


path  by  a  force  constantly  directed  towards  the  centre,  as 
when  a  solid  body  is  confined  by  a  string  and  whirled  around 
a  fixed  point.  In  this  case,  the  tangential  component  of  the 
deflecting  force  is  always  0.  There  will  be  no  loss  of  velo- 
city in  consequence  of  a  change  of  direction  in  the  motion 
(Art.  120).  Hence,  the  motion  of  the  point  will  be  uniform. 
Let  ABD  represent  the  path  of  the  body,  and  V  its 
centre.  Suppose  the  circumference 
of  the  circle  to  be  a  regular  polygon, 
having  an  infinite  number  of  sides,  of 
which  AB  is  one ;  and  denote  each 
of  these  sides  by  ds.  When  the  body 
reaches  A,  it  tends,  by  virtue  of  its 
inertia,  to  move  in  the  direction  of  the 
tangent  A  T ;  but,  in  consequence  of 
the  action  of  the  centripetal  force  di- 
rected towards  V,  it  is  constrained  to 
describe  the  side  ds  in  the  time  dt.    If 

we  draw  BO  parallel  to  AT,  it  will  be  perpendicular  to  the 
diameter  AD,  and  AC  will  represent  the  space  through 
which  the  body  has  been  drawn  from  the  tangent,  in  the 
time  dt.  If  we  denote  the  acceleration  due  to  the  centripetal 
force  by/,  and  suppose  it  to  be  constant  during  the  time  dt, 
we  shall  have,  from  Art.  114, 


AC=  \fde 


(122.) 


From  a  property  of  right-angled  triangles,  we  have,  since 
AB  =  ds, 

ds'  =  AC  X  AD  ;      or,     ds'  =  AC  X  2r. 
Whence, 


AC 


2r 


Substituting  this  value  of  AC  in  (122),  and  solving  with 
respect  to  /, 

J  ~  df       r 


CURVILINEAR    AND    ROTARY    MOTION.  199 

But  -^  =  v2  (Art.  113),  in  which  v  denotes  the  velocity 

of  the  moving  point.     Substituting  in  the  preceding  equa- 
tion, we  have, 

v2 

/=JT ("«•) 

Here  f  is  the  acceleration  due  to  the  deflecting  force; 
and,  since  this  is  exactly  equal  to  the  centrifugal  force,  we 
have  the  acceleration  clue  to  the  centrifugal  force  equal  to 
the  square  of  the  velocity,  divided  by  the  radius  of  the 
circle. 

If  the  mass  of  the  body  be  denoted  by  31,  and  the  entire 
centrifugal  force  by  F,  we  shall  have  (Art.  24), 

„      Mo* 

F  =  — - 124.) 

r 

If  we  suppose  the  body  to  be  moving  on  any  curve  what- 
ever, we  may,  whilst  it  is  passing  over  any  two  consecutive 
elements,  regard  it  as  moving  on  the  arc  of  the  oscillatory 
circle  to  the  curve  which  contains  these  elements  ;  and,  fur- 
ther, we  may  regard  the  velocity  as  uniform  during  the 
infinitely  small  time  required  to  describe  these  elements. 
The  direction  of  the  centrifugal  force  being  normal  to  the 
curve,  must  pass  through  the  centre  of  the  oscillatory  circle. 
Hence,  all  the  circumstances  of  motion  are  the  same  as 
before,  and  Equations  (123)  and  (124)  will  be  applicable, 
provided  r  be  taken  as  the  radius  of  the  curvature.  Hence, 
we  may  enunciate  the  law  of  the  centrifugal  force  as 
follows : 

The  acceleration  due  to  the  centrifugal  force  is  equal  to 
the  square  of  t/ie  velocity  of  the  body  divided  by  the  radius 
of  curvature. 

The  entire  centrifugal  force  is  equal  to  the  acceleration, 
multiplied  by  the  mass  of  the  body. 

In  the  case  of  a  body  whirled  around  a  centre,  and  re- 
strained by  a  string,  the  tension  of  the  string,  or  the  force 


' 


200 


MECHANICS. 


exerted  to  bicak  it,  will  be  measured  by  the  centrifugal 
force.  The  radius  remaining  constant,  the  tension  will 
increase  as  the  square  of  the  velocity. 

Centrifugal  Force  at  points  of  the  Earth's  Surface. 

136.  Let  it  be  required  to  determine  the  centrifugal 
force  at  different  points  of  the  earth's  surface,  due  to  ,ts 
rotation  on  its  axis. 

Suppose  the  earth  spherical.  Let  A  be  any  point  on  the 
surface,  PQP  a  meridian 
section  through  A,  PP  the 
axis,  FQ  the  equator,  and 
AB  perpendicular  to  PP\ 
the  radius  of  the  parallel  of 
latitude  through  A.  Denote 
the  radius  of  the  earth  by  r, 
the  radius  of  the  parallel 
through  A  by  r',  and  the 
latitude  of  A,  or  the  angle 
ACQ,  by  /.  The  time  of 
revolution  being  the  same  for  every  point  on  the  earth's 
surface,  the  velocities  of  Q  and  A  will  be  to  each  other  as 
their  distances  from  the  axis.  Denoting  these  velocities  by 
v  and  v',  we  have, 

v  :  v'  :  :  r  :  r\ 
whence, 

vr' 

v'  —  —  • 
r 


But,  from  the  right-angled  triangle  CAB,  since  the  angle 
at  A  is  equal  to  ",  we  have, 

r'  —  r  cosl. 


Substituting  this  value  of  r'  in   the  value  of  i/,  and  re- 
ducing, we  have, 

v'  =  v  cosl. 


CURVILINEAR    AND    ROTARY    MOTION.  201 

If  we  denote  the  acceleration  due  to  the  centrifugal  force 
at  the  equator  by/*  we  shall  have,  Equation  (123), 

v1 
f  =, (125.) 

In  like  manner,  if  we  denote  the  acceleration  due  to  the 
centrifugal  force  at  A,  by/',  we  shall  have, 


r 


Substituting  for  v'  and  r'  their  values,  previously  deduced, 
we  get, 

r=^L (126.) 

Comparing  Equations  (125)  and  (126),  we  find, 

/:/'::  1  :  cos?,  .-.    f  =/cos*    .    (127.) 

That  is,  the  centrifugal  force  at  any  point  on  the  earth's 
surface  is  equal  to  the  centrifugal  force  at  the  equator, 
multiplied  by  the  cosine  of  the  latitude  of  the  place. 

Let  AE,  perpendicular  to  PP\  represent  the  value  of 

/',  and  resolve  it  into  two  components,  one  tangential,  and 

the  other  normal  to  the  meridian  section.    Prolong  CA,  and 

draw  AD  perpendicular  to  it  at  A.     Complete  the  rectangle 

ED  on  AE  as  a  diagonal.     Then  will  AD  represent  the 

tangential,  and  AE  the  normal  component  of  /'.     In  the 

right-angled  triangle  AFE,  the  angle  at  A  is  equal  to  I. 

Hence, 

/sin  2? 
FE  =  AD  =  /'sin?  =  fcoslsinl  =  J—-—      .     ( 128.) 

AE  =  /'cos?  =  fcosH    .     .     .     .     ( 129.) 

From  (128),  we  conclude  that  the  tangential  component  is 

9* 


202  MECHANICS. 

0  at  the  equator,  goes  on  increasing  till  I  =  45°,  where  it 
is  a  maximum ;  then  goes  on  decreasing  till  the  latitude  is 
90°  when  it  again  becomes  0. 

The  effect  of  the  tangential  component  is  to  heap  up  the 
particles  of  the  earth  about  the  equator,  and,  were  the 
earth  in  a  fluid  state,  this  process  would  go  on  till  the  effect 
of  the  tangential  component  was  exactly  counterbalanced 
by  component  of  gravity  acting  down  the  inclined  plane 
thus  found,  when  the  particles  would  be  in  a  state  of  equili- 
brium. The  higher  analysis  has  shown  that  the  form  of 
equilibrium  is  that  of  an  oblate  spheroid,  differing  but 
slightly  from  that  which  our  globe  is  found  to  possess  by 
actual  measurement. 

From  Equation  (129),  we  see  that  the  normal  component 
of  the  centrifugal  force  is  equal  to  the  centrifugal  force  at 
the  equator  multiplied  by  the  square  of  the  cosine  of  the 
latitude  of  the  place. 

This  component  is  directly  opposed  to  gravity,  and,  con- 
sequently, tends  to  diminish  the  weight  of  all  bodies  on  the 
surface  of  the  earth.  The  value  of  this  component  is 
greatest  at  the  equator,  and  diminishes  towards  the  poles, 
where  it  becomes  equal  to  0.  From  the  action  of  the 
normal  component  of  the  centrifugal  force,  and  from  the 
flattened  form  of  the  earth  due  to  the  tangential  component 
bringing  the  polar  regions  nearer  the  centre  of  the  earth, 
the  measured  force  of  gravity  ought  to  increase  in  passing 
froni  the  equator  towards  the  poles.  This  is  found,  by 
observation,  to  be  the  case. 

The  radius  of  the  earth  at  the  equator  is  found,  by 
measurement,  to  be  about  3902.8  miles,  which,  multiplied  by 
2r,  will  give  the  entire  circumference  of  the  equator.  If 
this  be  divided  by  the  number  of  seconds  in  a  day,  86400, 
we  find  the  value  of  v.  Substituting  this  value  of  v  and 
that  of  r  just  given,  in  Equation  (125),  we  should  find, 

/  =  0.1112  ft., 
for  the  measure  of  the  centrifugal  force  at  the  equator.     If 


CURVILINEAR    AND    ROTARY    MOTION.  203 

this  be  multiplied  by  the  square  of  the  cosine  of  the  latitude 
of  any  place,  we  shall  have  the  value  of  the  normal  com- 
ponent  of  the  centrifugal  force  at  that  place. 

Centrifugal  Force  of  Extended  Masses. 

136.  We  have  supposed,  in  what  precedes,  the  dimen 
sions  of  the  body  under  consideration  to  be  extremely  small ; 
let  us  next  examine  the  case  of  a  body,  of  any  dimensions 
whatever,  constrained  to  revolve  about  a  fixed  axis,  with 
which  it  is  invariably  connected.  If  we  suppose  this  body 
to  be  divided  into  infinitely  small  elements,  whose  directions 
are  parallel  to  the  axis,  the  centrifugal  force  of  each  element 
will,  from  what  has  preceded,  be  equal  to  the  mass  of  the 
element  into  the  square  of  its  velocity,  divided  by  its  dis- 
tance from  the  axis.  If  a  plane  be  passed  through  the  cen- 
tre of  gravity  of  the  body,  perpendicular  to  the  axis,  we 
may,  without  impairing  the  generality  of  the  result,  suppose 
the  mass  of  each  element  to  be  concentrated  at  the  point  in 
which  this  plane  cuts  the  line  of  direction  of  the  element. 

Let  XCT  be  the  plane  through  the  centre  of  gravity  of 
the  body  perpendicular  to  the  axis  of 
revolution,   AB   the  section   cut  out  a^^ 

of  the  body,  or  the  projection  of  the 
body  on*  the  plane,  and  C  the  point 
in  which  it  cuts  the  axis.  Take  C  as 
the  origin  of  a  system  of  rectangular 
axes,  and  let   GX  be  the  axis  of  A",         c  X 

CY  the  axis  of   Y,  and  let  m  be  the  Fig.  121 

point  at  which  the  mass  of  one  of  these 

filaments  is  concentrated,  and  denote  that  mass  by  m.  De- 
note the  co-ordinates  of  m  by  x  and  y,  its  distance  from 
C  by  r,  and  its  velocity  by  v.  The  centrifugal  force  of  the 
mass  m  will  be  equal  to 

my2 


If  we  denote  the  angular  velocity  of  the  body  by  V\  the 


^04  MECHANICS. 

velocity  of  tne  point  m  will  be  equal  to  rl77,  which,  being 
substituted  in  the  expression  for  the  centrifugal  force  just 
deduced,  gives 

mrV7*. 

Let  this  force  be  resolved  into  two  components,  respec- 
tively parallel  to  the  axes  CX  and  CY.  We  shall  have, 
for  these  components,  the  expressions, 

mr  V'2cosm  CX,      and    mr  V*smm  CX. 

But  from  the  figure,  we  have, 

cosm  CX  =   - ,     and    sinm  CX  =    -  • 
r  r 

Substituting  these  values  in  the  preceding  expressions, 
and  reducing,  we  have,  for  the  two  components, 

mx  T7"'2,      and    my  F"'2. 

In  like  manner,  if  we  denote  the  masses  of  the  remaining 
filaments  by  m',  m",  <fcc,  the  co-ordinates  of  the  points  at 
which  they  are  cut  by  the  plane  XCY,  by  xr,  y' ;  x'\  y", 
&c,  their  distances  from  the  axis  by  ?•',  r",  <fec,  aqfi  resolve 
the  centrifugal  forces  into  components,  respectively  parallel 
to  the  axes,  we  shall  have,  since  V    remains  the  same, 

m*  x'  V'\        m'  y'  V*  ; 

m"x"V'\        m"y"V"  ; 

<fc&,  &G. 

if  we  denote  the  sum  of  the  components  in  the  direction 
oi'  the  axis  of  X  by  JST,  and  in  the  direction  of  the  axis 
of  Y  b)    Y",   we  shall  have, 

X  =  2 (ma)  V'\     and     Y  =  2  {my)  V'\ 


CURVILINEAR    AND    ROTARY    MOTION.  205 

If,  now,  we  denote  the  entire  mass  of  the  body,  by  J/, 
and  suppose  it  concentrated  at  its  centre  of  gravity  0, 
whose  co-ordinates  are  designated  by  xx,  and  yx,  and  whose 
distance  from  C  is  equal  to  rv  we  shall  have,  from  the 
principle  of  the  centre  of  gravity  (Art.  51), 

2(mx)  =  Jfoj,     and     2  (my)  =  Myx, 
Substituting  above,  we  have, 

X  =  MV'Xv      and     T  =  MV*yv 

t 
If  we  denote  the  resultant  of  all  the  centrifugal  forces, 

which  will  be  the  centrifugal  force  of  the  body,  by  it,  we 

shall  have, 


But  if  the  velocity  of  the  centre  of  gravity  be  denoted  by 
J7",  Ave  shall  have, 

V  =  Vr1;     or,   V  =  ~  ; 

which,  substituted  in  the  preceding  result,  gives,  for  the 
resultant, 

E=^- (130.) 


The  line  of  direction  of   ~R  is  made  known  by  the  equa- 
tions, 

X  m         ,         Y 


cosa  =  — ,      and    cos£> 


it,  therefore,  passes  through  the  centre  of  gravity  0. 

Hence,  we  conclude,  that  the  centrifugal  force  of  an  ex- 
tended mass,  constr  ined  to  revolve  about  a  fixed  axis,  with 
which  it  is  invariably  connected,  is  the  same  as  though  the 
entire  mass  were  concentrated  at  its  centre  of  gravity. 


206  MECHANICS. 


Pressure  on  the  Axis. 


137.  The  centrifugal  force,  passing  through  the  centre 
of  gravity  and  intersecting  the  axis,  will  exert  its  entire 
effect  in  creating  a  pressure  upon  the  axis  of  revolution. 
By  inspecting  the  equation, 

B  ^  MVnrv 

we  see  that  this  pressure  will  increase  with  the  mass,  the 
angular  velocity,  and  the  distance  of  the  centre  of  gravity 
from  the  axis.  When  the  last  distance  is  0,  that  is,  when 
ttie  axis  of  revolution  passes  through  the  centre  of  gravity, 
there  will  he  no  pressure  on  the  axis  arising  from  the  centri- 
fugal force,  no  matter  what  may  he  the  mass  of  the  body  or 
its  angular  velocity.  Such  is  the  case  of  the  earth  revolving 
on  its  axis. 

Principal  Axes. 

138.  Suppose  the  axis  about  which  a  body  revolves  to 
become  free,  so  that  the  body  can  move  in  any  direction. 
If  that  axis  be  not  one  of  symmetry,  it  will  be  pressed  un- 
equally in  different  directions  by  the  centrifugal  force,  and 
will  immediately  alter  its  position.  The  body  will  for  an 
instant  rotate  about  some  other  line,  which  will  immediately 
change  its  position,  giving  place  to  a  new  axis  of  rotation, 
which  will  instantly  change  its  position,  and  so  on,  until  an 
axis  is  reached  which  is  pressed  equally  in  all  directions  by 
the  centrifugal  forces  of  the  elements.  The  body  will  then 
continue  to  revolve  about  this  line,  by  virtue  of  its  inertia, 
until  the  revolution  is  destroyed  by  the  action  of  some 
extraneous  force.  Such  an  axis  is  called  a  principal  axis 
of  rotation.  Every  body  has  at  least  one  such  axis,  and 
may  have  more.  The  axis  of  a  cone  or  cylinder  is  a  prin- 
cipal axis  ;  any  diameter  of  a  sphere  is  &  principal  axis;  in 
short,  any  axis  of  symmetry  of  a  homogeneous  solid  is  a 
principal  axis.  The  shortest  axis  of  an  oblate  spheroid  is 
a  principal  axis;  and  it  is  found  by  observation  that  all  of 
the  planets  of  the  solar  system,  which  are  oblate  spheroids, 


CURVILINEAR    AND    ROTARY    MOTION.  207 

revolve  about  their  shorter  axes,  whatever  may  be  the  incli- 
nation of  these  axes  to  the  planes  of  their  orbits.  Were 
the  earth,  by  the  action  of  any  extraneous  force,  constrained 
to  revolve  about  some  other  axis  than  that  about  which  it  is 
found  to  revolve,  it  would,  as  soon  as  the  force  ceased  to 
act,  return  to  its  present  axis  of  rotation. 

Experimental  Illustrations. 

139.     The  principles  relating  to  the  centrifugal    force 
admit  of  experimental  illustration.     The  instrument  repre- 
sented in  the  figure,  may  be  employed  to  show  the  value  of 
the  centrifugal  force.     A  repre- 
sents a  vertical  axle  upon  which         bWttt^       "°i 

is  mounted  a  wheel  E,  commu-  \   '  \  *\\ 

nicatincr  with  a  train  of  wheel-  jjT7 

•  re- 

work, by  means  of  which   the  G       I     g 

axle  may  be  made  to   revolve  ,  j 

with  any  angular  velocity.     At  F    m 

the  upper  end  of  the  axle  is  a 

forked  branch  BC,  sustaining  a  stretched  wire.     D  and  E 

are  two  balls  which  are  pierced  by  the  wire,  and  are  free  to 

move  along  it.     Between  B  and  E  is  a  spiral  spring,  whose 

axis  coincides  with  the  wire. 

Immediately  below  the  spring,  on  the  horizontal  part  of 
the  fork,  is  a  scale  for  determining  the  distance  of  the  ball 
E,  from  the  axis,  and  for  measuring  the  degree  of  compres- 
sion of  the  spring.  Before  using  the  instrument,  the  force 
required  to  produce  any  degree  of  compression  of  the 
spring  is  determined  experimentally,  and  marked  on  the 
scale. 

If  now  a  motion  of  rotation  be  communicated  to  the  axis, 
the  ball  D  will  at  once  recede  to  (7,  but  the  ball  E  will  be 
restrained  by  the  spiral  spring.  As  the  velocity  of  rotation 
is  increased,  the  spring  will  be  compressed  more  and  more, 
and  the  ball  E,  will  approach  B.  By  a  suitable  arrange- 
ment of  the  wheelwork,  the  angular  velocity  of  the  axis 
corresponding  to  any  degree  of  compression  may  be  ascer- 


208  MECHANICS 

tained.  We  have  thus  all  the  data  necessary  to  a  verifier 
tion  of  the  law  of  the  centrifugal  force. 

If  a  vessel  of  water  be  made  to  revolve  about  a  vertical 
axis,  the  interior  particles  will  recede  from  the  axis  on 
account  of  the  centrifugal  force,  and  will  be  heaped  up  about 
the  sides  of  the  vessel,  imparting  a  concave  form  to  the 
upper  surface.  The  concavity  will  become  greater  as  the 
angular  velocity  is  increased. 

If  a  circular  hoop  of  flexible  metal  be  fastened  so  that 
one  of  its  diameters  shall  coincide  with  the  axis  of  a 
whirling  machine,  its  lower  point  being  fastened  to  the 
horizontal  beam,  and  a  motion  of  rotation  be  imparted,  the 
portions  of  the  hoop  farthest  from  the  axis  will  be  most 
affected  by  the  centrifugal  force,  and  the  hoop  will  be 
observed  to  assume  an  elliptical  form. 

If  a  sponge,  filled  with  water,  be  attached  to  one  of  the 
arms  of  a  whirling  machine,  and  a  motion  of  rotation  be 
imparted,  the  water  will  be  thrown  from  the  sponge.  This 
principle  has  been  made  use  of  in  a  machine  for  drying 
clothes.  An  annular  trough  of  copper  is  mounted  upon  an 
axis  by  means  of  radial  arms,  the  axis  being  connected  with 
a  train  of  wheelwork,  by  means  of  which  it  may  be  put  in 
motion.  The  outer  wall  is  pierced  with  holes  for  the  escape 
of  the  water,  and  a  lid  serves  to  confine  the  articles  to  be 
dried.  To  use  this  instrument,  the  linen,  after  being 
washed,  is  placed  in  the  annular  space,  and  a  rapid  motion 
of  rotation  imparted  to  the  machine.  The  linen  is  thrown, 
by  the  centrifugal  force,  against  the  outer  wall  of  the  instru- 
ment, and  the  water,  being  partially  squeezed  out,  and  par- 
tially thrown  off  by  the  centrifugal  force,  escapes  through 
the  holes  made  for  the  purpose.  Sometimes  as  many  as 
1,500  revolutions  per  minute  are  given  to  the  drying 
machine,  in  which  case,  the  drying  process  is  very  rapid  and 
very  perfect. 

If  a  body  be  whirled  about  an  axis  with  sufficient  velo- 
city, it  may  happen  that  the  centrifugal  force  generated 
will  be  greater  than  the  force  of  cohesion  which  binds  the 


CURVILINEAR    ANT)    ROTARY    MOTION.  209 

particles  together,  in  which  case,  the  body  will  be  torn 
asunder.  It  is  a  common  occurrence  that  large  grindstones, 
when  put  into  a  state  of  rapid  rotation,  burst,  the  fragments 
being  thrown  with  great  velocity  away  from  the  axis,  and 
often  producing  much  destruction. 

When  a  wagon,  or  carriage,  is  driven  rapidly  around  a 
corner,  or  is  forced  to  turn  about  a  circular  track,  the  cen- 
trifugal force  generated  is  often  sufficient  to  throw  out  the 
loose  articles  from  the  vehicle,  and  even  to  overthrow  the 
vehicle  itself.  When  a  car  upon  a  railroad  track  is  forced 
to  turn  around  a  sharp  curve,  the  centrifugal  force  generated, 
tends  to  throw  the  weight  of  the  cars  against  the  rail,  pro- 
ducing a  great  amount  of  friction,  and  contributing  to  wear 
out  both  the  track  and  the  car.  To  obviate  this  difficulty 
in  a  measure,  it  is  customary  to  raise  the  outer  rail,  so  that 
the  resultant  of  the  centrifugal  force,  and  the  force  of  grav- 
ity, shall  be  sensibly  perpendicular  to  the  plane  of  the  two 

rails. 

Elevation  of  the  outer  rail  of  a  curved  track. 

140.     To  find  the  inclination  of  the  track,  that  is,  the 
elevation  of  the  outer  rail,  so  that  the  resultant  of  the 
weight  and  centrifugal  force 
may  be  perpendicular  to  the  ^r 

line  joining  the  two  rails.  Let  x~"^X 

G  be  the  centre  of  gravity  j  \ 

of  the  car,  and  let  the  figure  1  \ 

represent    a   vertical   section  m        f[ ~^\\V\ 

through  the  centre  of  gravity  l|r^HJ \| 

and  the  centre  of  the  curved  ILi--- — * — -0 

track.     Let    GA9  parallel  to        E  Fi<r  V13  S 

the  horizon,  represent  the  ac- 
celeration due  to  the  centrifugal  force,  and  GJ3,  perpen- 
dicular to  the  horizon,  the  acceleration  due  to  the  weight 
of  the  car.  Construct  the  resultant  GC,  of  these  forces, 
then  must  the  line  DE\)Q,  perpendicular  to  GC.  Denote 
the  velocity  of  the  car,  by  v,  and  the  radius  of  the  curved 
track,  by  r.     The   acceleration  due  to  the  weight  will  be 


210  MECHANICS. 

equal  to  g,  the  force  of  gravity,  and  the  acceleration  due  to 

v2 
the  centrifugal  force  will  be  equal  to  —  •   The  tangent  of  the 

GB      r 

angle  GGB  will  be  equal  to  -p^=  ;    or,  denoting  the  angle 


a,  we  shall  have, 

GB 

V* 

tana  =  -  —  -  = 

GB 

gr 

But  the  angle  DEF  is  equal  to  the  angle  CGB.  Denot- 
ing the  distance  between  the  rails,  by  d,  and  the  elevation 
of  the  outer  rail  above  the  inner  one,  by  A,  we  shall  have, 

tana  =  — ,   very  nearly. 

Equating  the  two  values  of  tana,  we  have, 

h        v*  ,        dv*  # , «,  * 

j  =  — ,        .'.    h=  .     .     (131.) 

d       gr  gr  v 

Hence,  the  elevation  of  the  outer  rail  varies  as  the  square 
of  the  velocity  directly,  and  as  the  radius  of  the  curve 
inversely. 

It  is  obvious  that  this  correction  would  require  to  be 
different  for  different  velocities,  which,  from  the  nature  of 
the  case,  would  be  manifestly  impossible.  The  correction 
is,  therefore,  made  for  some  assumed  velocity,  and  then 
such  a  form  is  given  to  the  tire  of  the  wheels  as  will  com- 
plete the  correction  for  different  velocities. 

The  Conical  Pendulum. 

141.  The  conical  pendulum  consists  of  a  solid  ball  at- 
tached to  one  end  of  a  rod,  the  other  end  of  which  is  con- 
nected,  by  means  of  a  hinge-joint,  with  a  vertical  axle. 
When  the  axle  is  put  in  motion,  the  centrifugal  force  gene- 
rated in  the  ball  causes  it  to  recede  from  the  axis,  until  an 
equilibrium  is  established  between  the  weight  of  the  ball,  the 
centrifugal  force,  and   the  tension  of  the  connecting  rod. 


CURVILINEAR    AND    ROTARY    MOTION.  211 

When  the  velocity  is  constant,  the  centrifugal  force  will  be 
constant,  and  the  centre  of  the  ball  will  describe  a  horizontal 
circle,  whose  radius  will  depend  upon  the  velocity.  Let  it 
be  required  to  determine  the  time  of  revolution. 

Let  BD  be  the  vertical  axis,  A  the  ball,   B  the  hinge- 
joint,  and  AB  the  connecting  rod,  whose 
mass  is  so  small,  that  it  may  be  neglected, 
in  comparison  with  that  of  the  ball. 

Denote  the  required  time  of  revolution, 
by  t,  the  length  of  the  arm,  by  I,  the  accele- 
ration due  to  the  centrifugal  force,  by/*,  and 
the  angle  AB  (7,  by  p.  Draw  A  C  perpen- 
dicular to  BD,  and  denote  A  (7,  by  r,  and 
BC,hyh, 

From  the  triangle  AB  (7,  we  have,  r  =  Jsinp  ;  and  since 
r  is  the  radius  of  the  circle  described  by  A,  we  have  the 
distance  passed  over  by  A,  in  the  time  t,  equal  to 
2*r  =  2irlsinz>.  Denoting  the  velocity  of  A,  by  v,  we  have, 
from  Equation  (55), 

2<rrhmv 

v  =    - 


But   the  centrifugal   force   is   equal  to   the  square  of  the 
velocity,  divided  by  the  radius  ;  hence, 

/=    —^      ....      (132.) 

The  forces  which  act  upon  A,  are  the  centrifugal  force  in 
the  direction  AJF,  the  force  of  gravity  in  the  direction  A  G, 
and  the  tension  of  the  connecting  rod  in  the  direction  AB. 
In  order  that  the  ball  may  remain  at  an  invariable  distance 
from  the  axis,  these  three  forces  must  be  in  equilibrium. 
Hence  (Art.  35), 

g  :f  :  :  smBAF  :  :  sim£/l£; 

but,  smBAF '=  sin(90°  +  <p)  —  cos?; 


212  MECI1ANIC8. 

and,  sinBAG  —  sin(180°  —  <p)  =  sin?  ; 

whence,  by  substitution, 


ffifi:  cos?)  :  sinp,  ,\     g  =  f 


cos? 

sin  9 


Substituting  for  f  its  value,  taken  from  (132),  we  have, 

4<7r2£cosp 

9  =  —e — 

But,  from  the  triangle  A B  G,  we  have,   fcosp  =  h,   wl  ich 
gives, 

4*r«A  _       lh 


9  = 


P-  ' 


:2V-    '      ■      (1330 


That  is,  the  time  of  a  revolution  is  equal  to  the  time  of  a 
double  vibration  of  a  pendulum  whose  length  is  h. 


The  Governor. 

142.  The  principle  of  the  conical  pendulum  is  employed 
in  the  governor,  a  machine  attached  to  engines,  to  regulate 
the  motive  force. 

AB  is  a  vertical  axis  connected  with  the  machine  near  its 
working  point,  and  revolving  with  a 
velocity  proportional  to  that  of  the 
working  point ;  FE  and  GD  are  two 
arms  turning  freely  about  AB,  and 
bearing  heavy  balls  D  and  E,  at  their 
extremities;  these  bars  are  united  by 
hinge-joints  with  two  other  bars  at 
G  and  F,  these  bars  are  also  attach- 
ed to  a  ring  at  II,  free  to  slide  up  and 
down  the  shaft. 

The  governor  is  so  constructed, 
that  the  figure  GCFII  is  always  a  parallelogram.  The 
ring  at/Hs  connected  with  a  lever  HE,  which  maybe  made 
to  act  upon  the  valve  that  admits  steam  to  the  cylinder. 


Fig.  125. 


CURVILINEAR    AXD    ROTARY    MOTION.  213 

When  the  shaft  revolves,  the  centrifugal  force  developed 
in  the  balls,  causes  them  to  recede  from  the  axis,  and  the 
ring  H  is  depressed ;  and  when  the  velocity  has  become 
sufficiently  great,  the  lever  begins  to  act  in  closing  the  valve. 
If  the  velocity  slackens,  the  balls  approach  the  axis,  and  the 
ring  II  ascends,  opening  the  valve  again.  In  any  given 
case,  if  we  know  the  velocity  required  at  the  working  point, 
we  can  from  it  compute  the  required  angular  velocity  of  the 
shaft,  and,  consequently,  the  value  of  t.  This  value  of  t 
being  substituted  in  Equation  (133),  makes  known  the  value 
of  A.  We  may,  therefore,  make  the  proper  adaptation  of 
the  ring,  and  of  the  lever  UK. 

EXAMPLES. 

1.  A  ball  weighing  10  lbs.  is  whirled  around  in  a  circle 
whose  radius  is  10  feet,  with  a  velocity  of  30  feet  per  second. 
What  is  the  acceleration  of  the  centrifugal  force  ? 

A?is.  90  ft. 

2.  In  the  preceding  example,  what  is  the  tension  upon 
the  cord  which  restrains  the  ball  ? 

SOLUTION. 

Denote  the  tension  in  pounds,  by  t ;  then,  since  the  pres- 
sures produced  by  two  forces  are  proportional  to  their 
accelerations,  we  shall  have, 

10  :  t  :  :  g  :  90,  .-.     t  =  28  lbs.,  nearly.     Arts. 

3.  A  body  is  whirled  around  in  a  circular  path  whose 
radius  is  5  feet,  and  it  is  observed  that  the  pressure  due  to 
the  centrifugal  force  is  just  equal  to  the  weight  of  the  body. 
What  is  the  velocity  of  the  moving  body  ? 

SOLUTION. 

Denoting  the  velocity  by  t>,  we  have  the  acceleration 
due  to  the  centrifugal  force  equal  to  —  ;    but,  by  the  condi- 


214:  MECHANICS. 

tions  of  the  problem,  this  is  equal  to  the  acceleration  due  to 
the  weight  of  the  body.     Hence, 

%  =  g  =  321,  .-.     v  =  12.7  ft.     Am. 

5 

4.  In  how  many  seconds  must  the  earth  revolve  on  its 
axis  in  order  that  the  centrifugal  force  at  the  equator  may 
exactly  counterbalance  the  force  of  gravity,  the  radius  of 
the  equator  being  taken  equal  to  3962.8  miles  ? 

SOLUTION. 

Reducing  the  miles  to  feet,  and  denoting  the  required 
velocity,  by  v,  we  have, 


20923584 


=  321  ...     v  _  y^i  x  20923584. 


But  the  time  of  revolution  is  equal  to  the  circumference 
of  the  equator,  divided  by  the  velocity.  Denoting  the  time 
by  t,  we  have, 


_  2f  X  20923584 

v 


and,  substituting  the  value  of  v,  taken  from  the  preceding 
equation,  we  have,  after  reduction, 


2*V20923584        2tf  X  4574        ^nn 

t  =  — —— =  — =  5068  sees.     A?is. 

5.67 


But  the  earth  actually  revolves  in  86400  sideral,  or  in 
about  86164  mean  solar  seconds.  Hence,  the  earth  would 
have  to  revolve  1 7  times  as  fast  as  at  present,  in  order  that 
the  centrifugal  force  at  the  equator  might  be  equal  to  the 
force  of  gravity. 

5.  A  body  is  placed  on  a  horizontal  plane,  which  is 
made  to  revolve   about   a  vertical  axis,  with   an   angular 


CURVILINEAR    AND    ROTARY    MOTION.  215 

velocity  of  2  feet.  How  fur  must  the  body  be  situated  from 
the  axis  that  it  may  be  on  the  point  of  sliding  outwards,  the 
coefficient  of  friction  between  the  body  and  plane  being 
equal  to  .6  ? 

SOLUTION. 

Denote  the  required  distance  by  r ;  then  will  the  velocity 
of  the  body  be  equal  to  2r,  and  the  acceleration  due  to  the 
centrifugal  force  will  be  equal  to  4r.  But  the  acceleration 
due  to  the  force  of  friction  is  equal  to  0.6  x  g  —  19.3  ft. 
From  the  conditions  of  the  problem,  these  two  are  equal, 
hence, 

4r  =  19.3  ft.,  .'.     r  =  4.825  ft.     Ans. 

6.  What  must  be  the  elevation  of  the  outer  rail  of  a  rail- 
road track,  the  radius  of  curvature  being  3960  ft.,  the 
distance  between  the  rails  5  feet,  and  the  velocity  of  the  car 
30  miles  per  hour,  in  order  that  the  centrifugal  force  may 
be  exactly  counterbalanced  by  the  component  of  the  weight 
parallel  to  the  line  joining  the  rails  ? 

Ans.  0.076  ft.,  or  0.9  in.,  nearly. 

7.  The  distance  between  the  rails  is  5  feet,  the  radius  of 
the  curve  600  feet,  and  the  height  of  the  centre  of  gravity 
of  the  car  5  feet.  What  velocity  must  be  given  to  the  car 
that  it  may  be  on  the  point  of  being  overturned  by  the  cen- 
trifugal force,  the  rails  being  on  the  same  level  ? 

We  have, 

/o  x  321  x  600  -^         .-.  ,  A 

v  —  \/ - —  =  98  ft.,  or  66|  m.,  per  hour.    Ans. 

V  2x5 

Work. 

143.  By  the  term  work,  in  mechanics,  is  meant  the 
effect  produced  by  a  force  in  overcoming  a  resistance,  such 
as  weight,  inertia,  &c.  The  idea  of  work  implies  that  a 
force  is  continually  exerted,  and  that  the  point  at  which  it 
is  applied  moves  through  a  certain  space.  Thus,  when  a 
weight  is  raised  through  a  vertical  height,  the  p^wer  which 


216  MECHANICS. 

overcomes  the  resistance  offered  by  the  weight  is  said  to 
work,  and  the  amount  of  work  performed  evidently  depends, 
first,  upon  the  weight  raised,  and,  secondly,  upon  the 
height  through  which  it  is  raised.  All  kinds  of  work  may 
be  assimilated  to  the  raising  of  a  weight.  Hence  it  is,  that 
this  kind  of  work  is  assumed  as  a  standard  to  which  all 
other  kinds  of  work  are  referred. 

The  unit  of  work  most  generally  adopted  in  this  country, 
is  the  effort  required  to  raise  one  pound  through  a  height 
of  one  foot.  The  number  of  units  of  work  required  to  raise 
any  weight  to  any  height  will,  therefore,  be  equal  to  the 
product  obtained  by  multiplying  the  number  of  pounds  in 
the  weight  by  the  number  of  feet  in  the  height.  If  we 
take  the  weight  of  the  body  as  it  would  be  at  the  equator, 
for  the  sake  of  uniformity  in  notation,  we  may  regard  the 
weight  and  the  mass  as  identical  (Art.  11).  If  we  denote 
the  quantity  of  work  expended  in  raising  a  body,  by  Q,  the 
mass  of  the  body,  by  m,  and  the  height,  by  h,  we  shall  have, 

Q  —  mh. 

When  very  large  quantities  of  work  are  to  be  estimated, 
as  in  the  case  of  steam-engines  and  other  powerful  ma- 
chines, a  different  unit  is  sometimes  employed,  called  a 
horse  power.  When  this  unit  is  employed,  time  enters  as 
an  element.  A  horse  power  is  a  power  which  is  capable  of 
raising  33,000  lbs.  through  a  height  of  one  foot  in  one 
minute  ;  that  is,  it  is  a  power  capable  of  performing  33,000 
units  of  work  in  a  minute  of  time,  or  550  units  of  work  in 
one  second.  When  an  engine,  then,  is  spoken  of  as  being 
of  100  horse  power,  it  is  to  be  understood  that  it  is  capable 
of  performing  55,000  units  of  work  in  a  second. 

In  general,  if  a  force  acts  to  overcome  a  resistance  of  m 
pounds,  through  a  distance  of  n  feet,  whatever  may  be  the 
cause  of  the  resistance,  or  whatever  may  be  the  direction 
of  the  motion,  the  quantity  of  work  will  be  measured  by  a 
unit  of  work  taken  mn  times. 


CURVILINEAR    AND    ROTARY    MOTION.  217 

If  the  pressure  exerted  by  the  force  is  variable,  we  may 
conceive  the  path  described  by  the  point  of  application  to 
be  divided  into  equal  parts,  so  small  that,  for  each  part,  the 
pressure  may  be  regarded  as  constant.  If  we  denote  the 
length  of  one  of  these  equal  parts,  by  p,  and  the  force 
exerted  whilst  describing  this  path,  by  P,  we  shall  have  for 
the  corresponding  quantity  of  work,  Pp,  and  for  the  entire 
quantity  of  work  denoted  by  Q,  we  shall  have  the  sum  of 
these  elementary  quantities  of  work ;  or,  since  p  is  the 
same  for  each, 

Q=p2(P) (134.) 

The  quotient  obtained  by  dividing  the  entire  quantity  of 
work  by  the  entire  path,  is  called  the  mean  pressure,  or  the 
mean  resistance,  and  is  evidently  the  force  which,  acting 
uniformly  through  the  same  path,  would  accomplish  the 
same  work. 

Work,  when  the  power  acts  obliquely  to  the  path. 

144.     Let  PD  represent  the  force,  and  AB  the  path 
which  the  body  D  is  constrained  to 
follow.     Denote  the  angle  PDs  by  a, 
and  suppose  Pto  be  resolved  into  two 


components,   one   perpendicular,    and     ^        s  D       B 

the  other  parallel  to  AB.     We  shall 

have,  for  the  former,  7Jsina,  and,  for  the  latter,  Pcosa. 
The  former  can  produce  no  work,  since,  from  the  nature 
of  the  case,  the  point  cannot  move  in  the  direction  of  the 
normal  ;  hence,  the  latter  is  the  only  component  which 
works.  Let  sD  be  the  space  through  which  the  bodv  is 
moved  in  any  time  whatever.  If  we  denote  the  pressure 
exerted  in  the  direction  of  PD,  by  P,  and  the  quantity  of 
work,  by  Q,  we  shall  have, 

Q  —  Pcosa  x  s&. 

Let  fall  the  perpendicular  ss'  from  8,  on  the  direction  of  the 
10 


218  MECHANICS. 

force  P.      From   the    right-angled  triangle  Dss\  we  shall 
have, 

sD  x  cosa  =  s'D. 

Substituting  this  in  the  preceding  equation,  Ave  get, 

Q  =  P  x  s'D. 

That  is,  the  quantity  of  work  of  a  force  acting  obliquely 
to  the  path  along  which  the  point  of  application  is  con- 
strained to  move,  is  equal  to  the  intensity  of  the  force  mul- 
tiplied by  the  projection  of  the  path  upon  the  direction  of 
the  force.  We  have  supposed  the  intensity  of  the  force 
P,  to  be  expressed  in  pounds,  or  units  of  mass. 

If  we  take  the  distance  sD,  infinitely  small,  s'D  will  be 
the  virtual  velocity  of  Z>,  and  the  expression  for  the  quantity 
of  work  of  P  will  be  its  virtual  moment  (Art.  38).  Hence 
we  say  that  the  elementary  quantity  of  work  of  a  force  is 
equal  to  its  virtual  moment,  and,  from  the  principle  of 
virtual  moments,  we  conclude  that  the  algebraic  sum  of  the 
elementary  quantities  of  work  of  any  number  of  forces 
applied  at  the  same  point,  is  equal  to  the  elementary  quantity 
of  work  of  their  resultant.  What  is  true  for  the  elementary 
quantities  of  work  at  any  instant,  must  be  equally  true  at 
any  other  instant.  Hence,  the  algebraic  sum  of  all  the  ele- 
mentary quantities  of  work  of  the  components  in  any  time 
whatever,  is  equal  to  the  algebraic  sum  of  the  elementary 
quantities  of  work  of  their  resultant  for  the  same  time  ;  that 
is,  the  work  of  the  components  for  any  time,  is  equal  to  the 
work  of  their  resultant  for  the  same  time.  This  principle 
would  hardly  seem  to  require  demonstration,  for,  from  the 
very  definition  of  a  resultant,  it  would  seem  to  be  true  of 
necessity.  If  the  forces  are  in  equilibrium,  the  entire 
quantity  of  work  will  be  equal  to  0. 

This  principle  finds  an  important  application,  in  computing 
tin-  quantity  of  work  required  to  raise  the  material  for  a 
wall  or  building  ;  for  raising  the  material  from  a  shaft ;  for 
raising  water  from  one  reservoir  to  another  ;    and  a  great 


CURVILINEAR    AND    ROTARY    MOTION.  219 

variety  of  similar  operations.  In  this  connection,  the  prin- 
ciple may  be  enunciated  as  follows :  The  algebraic  sum  of 
the  quantities  of  work  required  to  raise  the  parts  of  a 
system  through  any  vertical  spaces,  is  equal  to  the  quantity 
of  work  required  to  move  the  whole  system  over  a  vertical 
space  equal  to  that  described  by  the  centre  of  gravity  of  the 
system. 

It  also  follows,  from  the  same  principle,-  that,  if  all  the 
pieces  of  a  machine  which  moves  without  friction  be  in 
equilibrium  in  all  positions,  under  the  action  of  zce/ghts 
suspended  from  different  parts  of  the  machine,  the  centre 
of  gravity  of  the  system  will  neither  ascend  nor  descend 
whilst  the  machine  is  in  motion. 


Work,  when  a  body  is  constrained  to  move  upon  a  curve. 

145.     Let  AB  represent  the  curve,  and  suppose  that  the 
force  is  so  taken  that  its  line  of  direction  shall 
always  pass  through   a  point  P.     Divide  the 
curve  into  elements  so  small  that  each  may  be  a2 

taken    as   a  straight   line,    and,  with  P  as  a  / . 

centre,  and  the  distances  from  P  to  the  points        A/ 
of  division  as  radii,  describe  arcs  of  circles. 
Then,  denoting  the  force  supposed  constant,  by  p 

P,  we   shall   have    (from  Art.    144)   the  ele-  Fig.  127. 

mentary  quantity  of  work  performed  whilst  the 
point  is  moving  over  aa',  equal  to  P  x  ac,  or  P  x  bb'.  In  like 
manner,  the  quantity  of  work  performed  whilst  the  point  is 
describing  a' a"  will  be  equal  to  P  x  b'b" ,  and  so  on.  Hence, 
by  summation,  we  shall  find  the  entire  quantity  of  work 
performed  in  moving  the  body  from  B  to  A  will  be  equal 
to  P  x  BB'.  If  now  we  suppose  the  curve  AB  to  lie  in 
a  vertical  plane,  and  the  force  to  be  the  force  of  gravity,  the 
point  P  may  be  regarded  as  infinitely  distant,  the  lines 
Pa,  Pa'  <fcc,  will  become  vertical,  and  the  lines  a'b',  a"b'\ 
will  be  horizontal.  We  may,  therefore,  enunciate  the  follow- 
ing principle  :  The  quantity  of  work  of  the  weight  of  a  body 


220  MKCHAN1C8. 

in  descending  a  curve,  is  equal  to  the  quantity  of  work  of 
tlie  same  weight  in  descending  vertically  through  the  same 
height.  This  principle  is  immediately  connected  with  the 
discussion  in  Art.  74. 

If  a  body  in  a  stable  position,  as  a  pyramid  resting  on  its 
base,  be  overturned  by  any  extraneous  force,  the  quantity 
of  work  will  be  equal  to  the  weight  of  the  body,  multiplied 
by  the  vertical  height  to  which  the  centre  of  gravity  must 
be  raised  before  reaching  its  highest  point.  This  product 
might  be  taken  as  the  measure  of  the  stability  of  a  body. 

EXAMPLES. 

1.  What  amount  of  work  is  required  to  raise  500  lbs.  to 
the  height  of  5  yards  ?  Ans.  7500  units,  or  7500  lbs.  ft. 

2.  To  what  height  can  2240  lbs.  be  raised  by  the  expen- 
diture of  5600  units  of  work?  Ans.  2.5  ft. 

3.  What  weight  can  be  raised  to  the  height  of  25  feet  by 
224000  units  of  work  ?  Ans.  8960  lbs. 

4.  What  is  the  effective  horse  power  of  an  engine  which 
raises  80  cubic  feet  of  water  per  minute  from  the  depth  of 
360  feet,  a  cubic  foot  of  water  weighing  62  lbs. 

Ans.  54.11  horse  power. 

5.  What  must  be  the  effective  horse  power  to  raise  the 
same  quantity  of  water  per  minute,  from  a  depth  of  40  feet  ? 

Ans.  6   horse  power. 

6.  How  many  tons  of  ore  can  be  raised  per  hour  from  a 
mine  1800  feet  deep,  by  an  engine  of  28  effective  horse 
power,  reckoning  2240  lbs.  to  the  ton?  Ans.   13J  tons. 

7.  From  what  depth  will  an  engine  of  16  effective  horse 
power  raise  5  cwts.  of  coal  per  minute. 

Ans.  943  feet,  nearly. 

8.  In  what  time  will  an  engine  of  40  effective  horse 
power  raise  44000  cubic  feet  of  water  from  a  mine  360  feet 
deep,  allowing  62i  pounds  to  the  cubic  foot? 

Ans.   12  h.  30  min. 


CURVILINEAR   AND    ROTARY    MOTION.  221 

9.  Required  the  quantity  of  work  necessary  to  raise  the 
material  for  a  rectangular  granite  wall  25  feet  long,  2\  feet 
thick,  and  20  feet  high,  the  weight  of  granite  being  162  lbs. 
per  cubic  foot  ? 

SOLUTION. 

The  weight  of  the  wall  is  equal  to 

162  lbs.  X  25  X  2.5  X  20  =  202500  lbs. 

The  height  of  the  centre  of  gravity  being  10  feet,  the 
quantity  of  work  is  equal  to 

202500  X  10  =  2025000  lbs.  ft.     Arts. 

10.  How  long  would  it  take  an  engine  of  4  effective 
horse  power  to  raise  the  material  for  the  wall  in  the  last 
example?  Ans.  \b\  minutes,  nearly. 

11.  What  quantity  of  work  must  be  expended  in  drawing 
a  chain  from  a  shaft,  the  length  of  the  chain  being  450  feet, 
and  its  weight  40  lbs.  to  the  foot  ?        Ans.  4050000  lbs.  ft. 

12.  A  cylindrical  well  is  150  feet  deep,  and  10  feet  in 
diameter.  Supposing  the  well  to  be  filled  with  water  to  the 
depth  of  50  feet,  how  much  work  must  be  expended  in 
raising  it  to  the  top,  water  being  taken  at  62.5  lbs.  per 
cubic  foot  ? 

SOLUTION. 

The  weight  of  the  water  is  equal  to 

*  X  52  X  50  X  62.5  lbs.  =  245437.5  lbs. 

The  distance  of  the  centre  of  gravity  from  the  top  is  125 
feet.     Hence,  the  required  quantity  of  work  is  equal  to 

245437.5  lbs.  X  125  ft.  =  30679687.5  lbs.  ft.     Ans. 

13.  What  quantity  of  work  will  be  required  to  overturn 
a  right  cone,  with  a  circular  base,  wrhose  altitude  is  12 


222  MECHANICS. 

feet,  and  the  radius  of  whose  base  is  4  feet,  the  weight  of 
the  material  being  estimated  at  100  lbs.  per  cubic  foot  ? 

SOLUTION. 

The  weight  of  the  cone  is  equal  to 

*  X  4a  X  4  X  100  lbs.  =  20106.24  lbs. 

If  the  cone  turns  about  a  tangent  to  its  base,  since  the 
centre  of  gravity  is  3  feet  from  the  base,  it  will  be,  * 


y's2  +  42  =  5  feet  from  the  tangent. 

The  centre  of  gravity,  at  its  highest  point,  will,  there- 
fore, be  5  feet  from  the  horizontal  plane.  It  must  then  be 
raised  2  feet.  Hence,  the  required  quantity  of  work  is 
equal  to 

20106.24  lbs.  X  2  ft.  =  40212.48  lbs.  ft.     Ans. 

14.  To  show  that  the  work  required  for  overturning 
similar  solids,  similarly  placed,  varies  as  the  fourth  powers 
of  their  homologous  lines. 

SOLUTION. 

Denote  the  altitudes  of  the  centres  of  gravity,  by  y  and  ry, 
the  distances  from  the  directions  of  the  weights  to  the  lines 
about  which  they  turn,  by  x  and  nc,  and  their  weights,  by 
to  and  raio. 

The  quantity  of  work  required  to  overturn  the  first, 
will  be, 

Q  =  tci^/x"  +  y*  -  y). 

The  quantity  of  work  required  to  overturn  the  second, 
will  be, 

Q'  =  r*w{y/r*xi  +  r*y*  —  ry)  =  r*w(^/x9  +  y*  -  y). 

Hence, 

Q      Q'  :  :  1  :  r*     .     .     Q.KD. 


CURVILINEAR    AND    ROTARY    MOTION.  223 

Rotation. 

146.  When  a  body  restrained  by  a  fixed  axis,  about 
which  it  is  free  to  turn,  is  acted  upon  by  a  force,  it  will,  in 
general,  take  up  a  motion  of  rotation,  or  revolution.  In 
this  kind  of  motion,  each  point  of  the  body  describes  a  cir- 
cle, whose  centre  is  in  the  axis,  and  whose  plane  is  perpen- 
dicular to  the  axis.  The  time  of  a  complete  revolution  be- 
ing the  same  for  each  particle,  it  follows,  that  the  velocities 
of  the  different  particles  will  be  proportional  to  their  dis- 
tances from  the  axis.  The  velocity  of  any  particle  will  be 
equal  to  its  distance  from  the  axis  multiplied  by  the  angular 
velocity  (Art,  122).  • 

Quantity  of  work  of  a  Force  producing  Rotation. 

147.  If  a  force  is  applied  obliquely  to  the  axis  of  rota- 
tion, we  may  conceive  it  to  be  resolved  into  two  components, 
one  parallel,  and  the  other  perpendicular  to  the  axis  of  rota- 
tion. The  effect  of  the  former  will  be  counteracted  by  the 
resistance  offered  by  the  fixed  axis  ;  the  effect  of  the  latter 
in  producing  rotation  will  be  exactly  the  same  as  that  of  the 
applied  force.  We  need,  therefore,  only  consider  those 
components  whose  directions  are  perpendicular  to  the  axis 
of  rotation. 

Let  P  represent  any  force  whose  line  of  direction  is  per- 
pendicular to  the  axis,  but  does 
not  intersect  it.     Let   0  be  the  D  _  f     _ 

point  in  which  a  plane  through  P,         \ Tf    ">''*-'£       *" 

perpendicular  to  the   axis,  inter-  // ',<>-"'" 

sects  it.     Let  A   and    C  be  any         j  fez-'' 

two  points  whatever,  on  the  line  FR128. 

of  direction  of  P.     Suppose  the 

force  P  to  turn  the  system  through  an  infinitely  small  angle, 

and  let  B  and  D  be  the  new  positions  of  A   and  C.     Draw 

OE,  Pa,  and  Dc  respectively  perpendicular  to  PE\  draw 

also,  A  0,  B  0,  CO,  and  Z>6>.     Denote  the  distances   OA, 

by  r,  00,  by  r',  OE,   by  p,  and  the  path   described  by 


224  MECHANICS. 

a  point  at  a  unit's  distance  from  0,  by  6'.     Since  the  angles 

A  OB,  and  COD  are  equal,  from 

the  nature  of  the  motion  of  rota-  r>  b 

E          **  C          'IP 
tion,   Ave    shall  have,  AB  =  rJ',       < -ff — -'^a. * 

and    CD  =  r'tJ'  ;  and   since   the  // s'l'-''' 

angular  motion  is  infinitely  small,       i  />--''' 

these  lines  may  be    regarded  as      ^ 

J  o  Fig.  128. 

straight  lines,    perpendicular   re- 
spectively to  OA  and  0  C     From  the  right-angled  triangles 
ABa  and  CDc,  we  have, 

.4a  =  rb'co&BAcLi      and     (7c  =  r'ycosDCc. 

In  the  right-angled  triangles  ABa,  and  OAE,  we  have 
^rl^>  perpendicular  to  0.4,  and  Aa  perpendicular  to  0E\ 
hence,  the  angles  BAa,  and  A  OE,  are  equal,  as  are  also 
their  cosines ;  hence,  we  have, 

cosBAa  =  cos  A  OE  =  £. 
r 

In  like  manner,  it  may  be  shown,  that 

cosDCc  =  cosCOE=  £• 
r 

Substituting  in  the  equations  just  deduced,  we  have, 
Aa  =  p$,       and     Cc  =  p&  ;        .-.     Aa  —  Cc  ; 

whence, 

P  .  Aa  =  P .  Cc  =  Dp  6'. 

The  iirst  member  of  the  equation  is  this  quantity  of  work 
of  P,  when  its  point  of  application  is  at  A  ;  the  second  is 
the  quantity  of  work  of  P,  when  its  point  of  application  is 
at  C.  Hence,  we  conclude,  that  the  elementary  quantity  of 
work  of  ei  force  applied  to  p/rocluce  rotation,  is  always  the 


CURVILINEAR    AND    ROTARY    MOTION.  225 

same,  wherever  its  point  of  application  may  be  taken,  pro- 
vided its  line  of  direction  remains  unchanged. 

We  conclude,  also,  that  the  elementary  quantity  of  work 
is  equal  to  the  intensity  of  the  force  multiplied  by  its  lever 
arm  into  the  elementary  space  described  by  a  point  at  a 
unit's  distance  from  the  axis. 

it'  we  suppose  the  force  to  act  for  a  unit  of  time,  the 
intensity  and  lever  arm  remaining  the  same,  and  denote  the 
angular  velocity,  by  t),  we  shall  have, 

Q'  =  Pp&- 

For  any  number  of  forces  similarly  applied,  we  shall  have, 
Q  =  2(Pp)6    .     .      .     .     (  135.) 

If  the  forces  are  in  equilibrium,  we  shall  have  (Art.  49), 
2(P/j>)  =  0;  consequently,    Q  =  0. 

Hence,  if  any  number  of  forces  tending  to  produce  rota- 
tion about  a  fixed  axis,  are  in  equilibrium,  the  entire  quan- 
tity of  work  of  the  system  of  forces  will  be  equal  to  0. 

Accumulation  of  Work. 

148.  When  a  body  is  put  in  motion  by  the  action  of  a 
force,  its  inertia  has  to  be  overcome,  and,  in  order  to  bring 
the  body  back  again  to  a  state  of  rest,  a  quantity  of  work 
has  to  be  given  out  just  equal  to  that  required  to  put  it  in 
motion.  This  results  from  the  nature  of  inertia.  A  body 
in  motion  may,  therefore,  be  regarded  as  the  representation 
of  a  quantity  of  work  which  can  be  reproduced  upon  any 
resistance  opposed  to  its  motion.  Whilst  one  body  is  in 
motion,  the  work  is  said  to  be  accumulated.  In  any  given 
instance,  the  accumulated  icork  depends,  first,  upon  the 
mass  in  motion ;  and,  secondly,  up^r  the  velocity  with  which 
it  moves. 

Take  the  case  of  a  body  y  jjected  vertically  upwards  in 
vacuum.  The  projecting  force  expends  upon  the  body  a 
quantity  of  work  sufficient  to  raise  it  through  a  height  equal 
10* 


L'26  MECHANIC8. 

to  that  due  to  the  velocity  of  projection.  Denoting  the 
weight  of  the  body,  by  w,  the  height  to  which  it  rises,  by  A, 
md  the  accumulated  work,  by  §,  we  shall  have, 

Q  =.  wh. 

i>2 
But,  h  —  J  — ,  (Art.  116),  hence, 

if 


Denoting  the    mass   of   the  body  by  m,   we   shall   have, 

10 
m  —  —    (Art.  11),  and,  by  substitution,  we  have,  finally, 

y 

Q  =  \mtf (  136.) 

If  the  body  descends  by  its  own  weight,  it  will  have 
impressed  upon  it  by  the  force  of  gravity,  during  the 
descent,  exactly  the  same  quantity  of  work  as  it  gave  out 
in  ascending. 

The  amount  of  work  accumulated  in  a  body  is  evidently 
the  same,  whatever  may  have  been  the  circumstances  under 
which  the  velocity  has  been  acquired ;  and  also,  the  amount 
of  work  which  it  is  capable  of  giving  out  in  overcoming  any 
resistance  is  the  same,  whatever  may  be  the  nature  of  that 
resistance.  Hence,  the  measure  of  the  accumulated  work 
of  a  moving  mass  is  one-half  of  the  mass  into  the  square 
of  the  velocity. 

The  expression  mv',  is  called  the  living  force  of  the 
body.  Hence,  the  living  force  of  a  body  is  equal  to  its 
mass,  multiplied  by  the  square  of  its  velocity.  The  living 
force  of  a  body  is  the  measure  of  twice  the  quantity  of 
work  expended  in  producing  the  velocity,  or,  it  is  the 
measure  of  twice  the  quantity  of  work  which  the  body  is 
capable  <>t'  giving  out. 

When  the  forces  exerted  tend  to  increase  the  velocity, 


CURVILINEAR    AND    ROTARY    MOTION.  227 

their  work  is  regarded  as  positive  ;  when  they  tend  to  dimin- 
ish it,  their  work  is  regarded  as  negative.  It  is  the  aggre- 
gate of  all  the  work  expended,  both  positive  and  negative, 
that  is  measured  by  the  quantity,  i/ny2. 

I±^  at  any  instant,  a  body  whose  mass  is  m,  has  a  velocity 
v,  and,  at  any  subsequent  instant,  its  velocity  has  become  v\ 
we  shall  have,  for  the  accumulated  work  at  these  two 
instants, 

Q  =  iray2,      Q'  =  ±mv'* ; 

and,  for  the  aggregate  quantity  of  work  expended  in  the 
interval, 

Q"  =  ±m(v'*  -  v")  .     .     .     .     (13V.) 

When  the  motive  forces,  during  the  interval,  perform  a 
greater  quantity  of  work  than  the  resistances,  the  value  of 
v'  will  be  greater  than  that  of  v,  and  there  will  be  an  accu- 
mulation of  work  in  the  interval.  When  the  work  of  the 
resistances  exceeds  that  of  the  motive  forces,  the  value  of  v 
will  exceed  that  of  v\  Q"  will  be  negative,  and  there  will 
be  a  loss  of  living  force,  which  is  absorbed  by  the  resistances. 

Living  Force  of  Revolving  Bodies. 

149.  Denote  the  angular  velocity  of  a  body  which  is 
restrained  by  an  axis,  by  d  ;  denote  the  masses  of  its  ele- 
mentary particles  by  m,  m\  &c,  and  their  distances  from 
the  axis  of  rotation,  by  r,  r',  &c.  Their  velocities  will  be 
?\\  r'A,  ifcc,  and  their  living  forces  will  be  mr'd5,  mV'Jf)J,  &c. 
Denoting  the  entire  living  force  of  the  body,  by  X,  Ave  shall 
have,  by  summation,  and  recollecting  that  $*  is  the  same  for 
all  the  terms, 

L  =  2(mry      ....     (138.) 

But  2(mr2)  is  the  expression  for  the  moment  of  inertia  of 
the  body,  taken  with  respect  to  the  axis  of  rotation.     De- 


22S  MECHANICS. 

noting  the  entire  mass  by  Jff,  its  radius  of  gyration,  with 
respect  to  the  axis  of  rotation,  by  k,  we  shall  have, 

L  =  MW. 

If,  at  any  subsequent  instant,  the  angular  velocity  aas 
become  d\  we  shall,  at  that  instant,  have, 

L'  =  MM"  ; 

and,  for  the  loss  or  gain  of  living  force  in  the  interval,  we 
shall  have, 

L"  =  MJP(6'*  —  6*).     .     .     (139.) 

If  we  make  &'*  —  ^  =  1,  we  shall  have, 

L"'  =  Mk>  -  2(mr2)    .     .     (140.) 

which  shows  that  the  moment  of  inertia  of  a  body,  with 
respect  to  an  axis,  is  equal  to  the  living  force  lost  oi 
gained  whilst  the  body  is  experiencing  a  change  in  the 
square  of  its  angular  velocity  equal  to  1. 

The  principle  of  living  forces  is  extensively  applied  in 
discussing  the  circumstances  of  motion  of  machines.  When 
the  motive  power  performs  a  quantity  of  work  greater  than 
that  necessary  to  overcome  the  resistances,  the  velocities  of 
the  parts  become  accelerated,  a  quantity  of  work  is  stored 
up,  to  be  again  given  out  when  the  resistances  offered 
require  a  greater  quantity  of  work  to  overcome  them  than 
is  furnished  by  the  motor. 

In  many  machines,  pieces  are  expressly  introduced  to 
equalize  the  motion,  and  this  is  particularly  the  case  when 
either  the  motive  power  or  the  resistance  to  be  overcome, 
is,  in  its  nature,  variable.     Such  pieces  are  called  fly-wheels. 

Fly-Wheels. 

150.  A  fly-wheel  is  a  heavy  wheel,  usually  of  iron, 
mounted  upon  an  axis}  near  the  point  of  application  of  the 


CURVILINEAR    AND    ROTARY    MOTION.  229 

force  which  it  is  destined  to  regulate.  It  is  generally  com- 
posed  of  a  heavy  rim,  connected  with 
the  axis  by  means  of  radial  arms. 
Sometimes  it  consists  of  radiating 
bars,  carrying  heavy  spheres  of  metal 
at  their  outer  extremity.  In  either 
case,  we  see,  from  Equation  139,  that, 
for  a  given  quantity  of  work  absorbed, 
the  value  of  d'-  —  oa  will  be  less  as  M 
and  k  are  greater  ;  that  is,  the  change  FisTi$» 

of  angular  velocity  will  be  less,  as  the 
mass  of  the  fly-wheel  and  its  radius  of  gyration  increase. 
It  is  for  this  reason  that  the  peculiar  form  of  fly-wheel 
indicated  above,  is  adopted,  it  being  the  form  that  most 
nearly  realizes  the  conditions  pointed  out.  The  principal 
objection  to  large  fly-wheels  in  machinery,  is  the  great 
amount  of  hurtful  resistance  which  they  create,  such  as  fric- 
tion on  the  axle,  etc.  Thus,  a  fly-wheel  of  42000  lbs.  would 
create  a  force  of  friction  of  4200  lbs.,  the  coefficient  of  fric- 
tion being  but  T\> ;  and,  if  the  diameter  of  the  axle  were 
8  inches,  and  the  number  of  revolutions  30  per  minute,  this 
resistance  alone  would  be  equal  to  8  horse  powers. 

EXAMPLES. 

1.  The  weight  of  the  ram  of  a  pile-driver  is  400  lbs.,  and 
it  strikes  the  head  of  a  pile  with  a  velocity  of  20  feet. 
What  is  the  amount  of  work  stored  up  in  it  ? 

SOLUTION. 

The  height  due  to  the  velocity,  20  feet,  is  equal  to 
—2.  -  6.22  ft.,  nearly. 

Hence,  the  stored  up  work  is  equal  to 

400  lbs.  X  6.22  ft.  =  2488  lbs.  ft. ; 


230  MECHANICS. 

or,  the  stored  up  work,  equal  to  half  the  living  force,  is 
equal  to 

400       (20)a 

— -  x  - — —  =  2488  units.     Ans. 

32i  2 

2.  A  train,  weighing  GO  tons,  has  a  velocity  of  40  miles 
per  hour  when  the  steam  is  shut  off.  How  far  will  it  travel, 
if -no  brake  be  applied,  before  the  velocity  is  reduced  to  10 
miles  per  hour,  the  resistance  to  motion  being  estimated  at 
10  lbs.  per  ton.  Ans.  '1 1236  ft. 

Composition  of  Rotations. 

151.  Let  a  body  A  CBD,  that  is  free  to  move,  be  acted 
upon  by  a  force  which,  of  itself, 
would  cause  the  body  to  revolve 
for  the  infinitely  small  time  dty 
about  the  line  AP,  with  an  angu- 
lar velocity  v ;  and  at  the  same 
instant,  let  the  body  be  acted 
upon  by  a  second  force,  which 
would  of  itself  cause  the  body  to 
revolve  about  CD,  for  the  time 
dt,  with  an   angular  velocity  v'. 

Suppose  the  axes  to  intersect  each  other  at  O,  and  let  P  be 
any  point  in  the  plane  of  the  axes.  Draw  PF  and  PG  res- 
pectively perpendicular  to  OP  and  OC\  denoting  the  for- 
mer, by  x,  and  the  latter,  by  y.  Then  will  the  velocity  of 
P  due  to  the  first  force,  be  equal  to  vx,  and  its  velocity  due 
to  the  second  force  will  be  equal  to  v'y.  Suppose  the  rota- 
tion to  take  place  in  such  a  manner,  that  the  tendency  of 
the  rotation  about  one  of  the  axes,  shall  be  to  depress  the 
point  below  the  plane,  whilst  that  about  the  other  is  to 
elevate  it  above  the  plane ;  then  will  the  effective  velocity 
of  P  be  equal  to  vx  —  v'y.  If  this  effective  velocity  is  0, 
the.  j)oi?it  P  icill  remain  at  rest.  Placing  the  expression 
ju*t  deduced  equal  to  0,  and  transposing,  we  have, 

vx  —  v'y. 


CURVILINEAR    AND    ROTARY    MOTION.  231 

To  determine  the  position  of  P,  lay  off  Off  equal  to  », 
01  equal  to  v',  and  regard  these  lines  as  the  representatives 
of  two  forces ;  we  have,  from  the  equation,  the  moment  of 
v,  with  respect  to  the  point  P,  equal  to  the  moment  of  v\ 
with  respect  to  the  same  point.  Hence,  the  point  P  must 
be  somewhere  upon  the  diagonal  Off,  of  the  parallelogram 
described  on  w,  and  v' .  But  P  may  be  anywhere  on  this 
line ;  hence,  every  point  of  the  diagonal  OK,  remains  at 
rest  during  the  time  dt,  and  is,  consequently,  the  resultant 
axis  of  rotation.  We  have,  therefore,  the  following  principles : 

If  a  body  be  acted  upon  simultaneously  by  two  forces, 
each  tending  to  impart  a  motion  of  rotation  about  a  sepa- 
rate axis,  the  resultant  motion  tcill  be  one  of  rotation  about 
a  third  axis  lying  in  the  plane  of 'the  other  tico,  and  passing 
through  their  common  point  of  intersection. 

The  direction  of  the  resultant  axis  coincides  with  the 
diagonal  of  a  parallelogram,  \ohose  adjacent  sides  are  the 
component  axes,  and  ichose  lengths  are  proportional  to  the 
impressed  angidar  velocities. 

Let  OH  and  01  represent,  as  before,  the  angular  veloci- 
ties v  and  v',  and  Off  the  diagonal  of  the 

parallelogram  constructed  on  these  lines  I "K 

as  sides.     Take  any  point  I,  on  the  second  /i^i^s 

axis,  and  let  fall  a  perpendicular  on  Off  and.       J^\       / 
Off;  denote  the  former  by  r,   and   the      0  H 

latter,  by  r"  ;  denote,   also,  the  resultant  Fig-  18L 

angular  velocity,  by  v".  Since  the  actual  space  passed  over 
by  I,  during  the  time  t,  depends  only  upon  the  first  force,  it 
will  be  the  same  whether  we  regard  the  revolution  as  taking 
place  about  the  axis  Off,  or  about  the  axis  Off.  If  we 
suppose  the  rotation  to  take  place  about  Off,  the  space 
passed  over  in  the  time  dt,  will  be  equal  to  rvdt ;  if  we  sup- 
pose the  rotation  to  take  place  about  Off,  the  space  passed 
over  in  the  same  time  will  be  equal  to  r"v"dt.  Placing 
these  expressions  equal  to  each  other,  we  have,  after  reduc- 
tion, 

r' 


232 


MECHANICS. 


But  regarding  I  as  a  centre  of  moments,  we  shall  hare, 
from  the  principle  of  moments, 

r 


OK  x  r"  =  vr :     or,      OK 


v. 


By  comparing  the  last  two  equations,  we  have, 
v"  =  OK. 

That  is,  the  resultant  angular  velocity  will  be  equal  to  the 
diagonal  of  the  parallelogram  described  on  the  component 
angular  velocities  as  sides. 

By  a  course  of  reasoning  entirely  similar  to  that  employed 
in  demonstrating  the  parallelopipedon  of  forces,  we  might 
show,  that, 

If  a  body  be  acted  upon  by  three  simultaneous  forces, 
each  tending  to  produce  rotation  about  separate  axes  inter- 
secting each  other,  the  resultant  motion  will  be  one  of  rota- 
tion about  the  diagonal  of  the  parallelopipedon  whose  adja- 
cent edges  are  the  component  angular  velocities,  and  the 
resultant  angular  velocity  will  be  represented  by  the  length 
of  this  diagonal. 

The  principles  just  deduced  are  called,  respectively,  the 
parallelogram  and  the  parallelopipedon  of  rotations. 


Application  to  the  Gyroscope. 

152.     The  gyroscope  is  an  instrument  used  to  illustrate 
the  laws  of  rotary  motion.     It  consists  essentially  of  a  heavy 
wheel  A,  mounted  upon 
an   axle  BC     This  axle 
is  attached,  by  means  of 
pivots,  to  the  inner  ed<_e 
of  a  circular   hoop  I)  1\ 
within   which    the  wheel 
A    can    t iii-n    freely.     On 
one  side  of  the  hoop,  and  in  the  prolongation  of  the  axle 
BC,  is  a  bar  EF,  having  a  conical  hole  drilled  on  its  lower 


Fig.  132. 


CTTRVILIMsAR    AND    ROTARY    MOTION.  233 

face  to  receive  the  pointed  summit  of  a  vertical  standard  G. 
If  a  string  be  wrapped  several  times  around  the  axle  J>C, 
and  then  rapidly  unwound,  so  as  to  impart  a  rapid  motion 
of  rotation  to  the  wheel  A,  in  the  direction  indicated  by 
the  arrow-head,  it  is  observed  that  the  machine,  instead  of 
sinking  downwards  under  the  action  of  gravity,  takes  up  a 
retrograde  orbital  motion  about  the  pivot  6r,  as  indicated  by 
the  arrow-head  IT.  For  a  time,  the  orbital  motion  in- 
creases, and,  under  certain  circumstances,  the  bar  EF  is 
observed  to  rise  upwards  in  a  retrograde  spiral  direction; 
and,  if  the  cavity  for  receiving  the  pivot  is  pretty  shallow. 
the  bar  may  even  be  thrown  off  the  vertical  standard. 
Instead  of  a  bar  EF,  the  instrument  may  simply  have  an 
ear  at  E,  and  be  suspended  from  a  point  above  by  means  of 
a  string  attached  to  the  ear.  The  phenomena  observed  are 
the  same  as  before. 

Before  explaining  these  phenomena,  it  will  be  necessary 
to  point  out  the  conventional  rules  for  attributing  proper 
signs  to  the  different  rotations. 

Let  OX,  OY,  and    OZ,  be  three  rectangular  axes.     It 
has  been  agreed  to  call  all  dis- 
tances,   estimated   from    0,  to- 
wards either  JT,  Y,  or  Z,  posi- 
tive /  consequently,  all  distances  0 
estimated  in  a  contrary  direction           V^l-  ss^i-^--' 
must  be  regarded  as  negative.       y     c 
If  a  body  revolve  about  either  Fig.  133. 
axis,  or  about  any  line  through 

the  origin,  in  such  a  manner  as  to  appear  to  an  eye  beyond 
it,  in  the  axis  and  looking  towards  the  origin,  to  move  in 
the  same  direction  as  the  hands  of  a  watch,  that  rotation  is 
considered  positive.  If  rotation  takes  place  in  an  opposite 
direction,  it  is  negative.  The  arrow-head  A,  indicates  the 
direction  of  positive  rotation  about  the  axis  of  Jl.  To  an 
eye  situated  beyond  the  body,  as  at  JT,  and  looking  towards 
the  origin,  the  motion  appears  to  be  in  the  same  direction 
as  the  motion  of  the  hands  of  a  watch.     The  arrowhead  B< 


3         P.'M 


11' 


234  MECHANICS. 

indicates  the  direction  of  positive  rotation  about  the  axis 
of  F,  and  the  arrow-head  C,  the  direction  of  positive  rota- 
tion about  the  axis  of  Z. 

Suppose  the  axis  of  the  wheel  of  the  gyroscope  to  coincide 
with  the  axis  of  A",  taken  horizontal;  let  the  standard  be 
taken  to  coincide  with  the  axis  of  Z,  the  axis  of  Y  bein^ 
perpendicular  to  them  both.  Let  a  positive  rotation  be 
communicated  to  the  wheel  by  means  of  a  string.  For  a 
very  short  time  dt,  the  angular  velocity  may  be  regarded 
as  constant.  In  the  same  time  dt,  the  force  of  gravity  acts 
to  impart  a  motion  of  positive  rotation  to  the  whole  instru- 
ment about  the  axis  of  Y,  which  may,  for  an  instant,  be 
regarded  as  constant.  Denote  the  former  angular  velocity 
by  tf,  and  the  latter  by  v'.  Lay  off  in  a  positive  direction 
on  the  axis  of  A,  the  distance  OD  equal  to  v,  and,  on  the 
positive  direction  of  the  axis  of  Y,  the  distance  OP  equal 
to  v',  and  complete  the  parallelogram  OF.  Then  (Art.  151) 
will  OF  represent  the  direction  of  the  resultant  axis  of  revo- 
lution, and  the  distance  OF  will  represent  the  resultant 
angular  velocity,  which  denote  by  v".  In  moving  from  OB 
to  OF,  the  axis  takes  up  a  positive,  or  retrograde  orbital 
motion  about  the  axis  of  Z.  To  construct  the  position  of 
the  resultant  axis  for  the  second  instant  dt,  we  must  com- 
pound three  angular  velocities.  Lay  off  on  a  perpendicular 
to  OF  and  OZ,  the  angular  velocity  OG  due  to  the  action 
of  gravity  during  the  time  dt,  and  on  OZ  the  angular  velo- 
city in  the  orbit ;  construct  a  parallelopipedon  on  these 
lines,  and  draw  its  diagonal  through  O.  This  diagonal 
will  coincide  in  direction  with  the  resultant  axis  for  the 
second  instant,  and  its  length  will  represent  the  resultant 
angular  velocity  (Art.  151).  For  the  next  instant,  we  may 
proceed  as  before,  and  so  on  continually.  Since,  in  each 
case,  the  diagonal  is  greater  than  either  edge  of  the  paral- 
lelopipedon, it  follows  that  the  angular  velocity  will  contin- 
ually increase,  and,  were  there  no  hurtful  resistances,  this 
increase  would  go  on  indefinitely.  The  effect  of  gravity  is 
continually  exerted  to  depress  the  centre  of  gravity  of  the 


CURVILINEAR    AND    ROTARY    MOTION".  235 

instrument,  whilst  the  effect  of  the  orbital  rotation  is  to 
elevate  it.  When  the  latter  effect  prevails,  the  axis  of  the 
gyroscope  will  continually  rise ;  when  the  former  prevails, 
the  gyroscope  will  continually  descend.  Whether  the  one 
or  the  other  of  these  conditions  will  be  fulfilled,  depends 
upon  the  angular  velocity  of  the  wheel  of  the  gyroscope, 
and  upon  the  position  of  the  centre  of  gravity  of  the  instru- 
ment. Were  the  instrument  counterpoised  so  that  the 
centre  of  gravity  would  lie  exactly  over  the  pivot,  there 
would  be  no  orbital  motion,  neither  would  the  instrument 
rise  or  fall.  Were  the  centre  of  gravity  thrown  on  the 
opposite  side  of  the  pivot  from  the  wheel,  the  rotation  due 
to  gravity  would  be  negative,  that  is,  the  orbital  motion 
would  be  direct,  instead  of  retrograde. 


236  MECHANICS 


CHAPTER     VII. 

MECHANICS      OF     LIQUIDS. 

Classification  of  Fluids. 

153.  A  fluid  is  a  body  whose  particles  move  freely 
amongst  each  other,  each  particle  yielding  to  the  slightest 
force.  Fluids  are  of  two  classes  :  liquids,  of  which  water  is 
a  type,  and  gases,  or  vapors,  of  which  air  and  steam  are 
types.  The  distinctive  property  of  the  first  class  is,  that 
they  are  sensibly  incompressible;  thus,  water,  on  being 
pressed  by  a  force  of  15  lbs.  on  each  square  inch  of  surface, 
only  suffers  a  diminution  of  about  yooVo o  °f  ^ts  Dulk.  The 
second  class  comprises  those  which  are  readily  compressible ; 
thus,  air  and  steam  are  easily  compressed  into  smaller  vol- 
umes, and  when  the  pressure  is  removed,  they  expand,  so  as 
to  occupy  larger  volumes. 

Most  liquids  are  imperfect ;  that  is,  there  is  more  or  less 
adherence  between  their  particles,  giving  rise  to  viscosity. 
In  what  follows,  they  will  be  regarded  as  destitute  of  vis- 
cosity, and  homogeneous.  For  certain  purposes,  fluids  may 
also  be  regarded  as  destitute  of  weight,  without  impairing 
the  validity  of  the  conclusions. 

Principle  of  Equal  Pressures. 

154.  From  the  nature  and  constitution  of  a  fluid,  it  fol- 
lows, that  each  of  its  particles  is  perfectly  movable  in  all 
directions.  From  this  fact,  we  deduce  the  following  funda- 
mental law,  viz. :  If  a  fluid  is  in  equilibrium  under  the 
action  of  any  forces  whatever,  each  particle  of  the  mass  is 
equally  pressed  in  all  directions  /  for,  if  any  particle  were 
more  strongly  pressed  in  one  direction  than  in  the  others, 


MECHANICS    OF    LIQUIDS.  237 

it  would  yield  in  that  direction,  and  motion  it  ould  ensue, 
which  is  contrary  to  the  hypothesis. 

This  is  called  the  principle  of  equal  pressures. 

It  follows,  from  the  principle  of  equal  pressures,  that  if 
any  point  of  a  fluid  in  equilibrium,  be  pressed  by  any  force, 
that  pressure  will  be  transmitted  without  change  of  intensity 
to  every  other  point  of  the  fluid  mass. 

This  may  be  illustrated  experimentally,  as  follows: 

Let  AB  represent  a  vessel  filled  with  a  fluid  in  equili- 
brium. Let  C  and  D  represent  two 
openings,  furnished  with  tightly-fit- 
ting pistons.  Suppose  that  forces  are 
applied  to  the  pistons  just  sufficient  to 
maintain  the  fluid  mass  in  equilibrium. 
If,  now,  any  additional  force  be  appli- 
ed to  the  piston  P,  the  piston  Q  will 
be  forced  outwards ;  and  in  order  to 
prevent  this,  and  restore  the  equili- 
brium, it  will  be  found  necessary  to  apply  a  force  to  the 
piston  g,  which  shall  have  the  same  ratio  to  the  force  ap- 
plied at  P  that  the  area  of  the  piston  Q  has  to  the  area  of 
the  piston  P.  This  principle  will  be  found  to  hold  true, 
whatever  may  be  the  sizes  of  the  two  pistons,  or  in  what- 
ever portions  of  the  surface  they  may  be  inserted.  If  the 
area  of  P  be  taken  as  a  unit,  then  will  the  pressure  upon  Q 
be  equal  to  the  pressure  on  P,  multiplied  by  the  area  of  Q. 

The  pressure  transmitted  through  a  fluid  in  equilibrium, 
to  the  surface  of  the  containing  vessel,  is  normal  to  that  sur- 
face ;  for  if  it  were  not,  we  might  resolve  it  into  two  compo- 
nents, one  normal  to  the  surface,  and  the  other  tangential ; 
the  effect  of  the  former  would  be  destroyed  by  the  resistance 
of  the  vessel,  whilst  the  latter  would  impart  motion  to  the 
fluid,  which  is  contrary  to  the  supposition  of  equilibrium. 

In  like  manner,  it  may  be  shown,  that  the  resultant  of  al1 
the  pressures,  acting  at  any  point  of  the  free  surface  of  a 
fluid,  is  normal  to  the  surface  at  that  point.  When  the  only 
force  acting  is  the  force  of  gravity,  the  surface  is  level.     For 


238  MECHANICS. 

small  areas,  a  level  surface  coincides  sensibly  with  a  horizon- 
tal  plane.  For  larger  areas,  as  lakes  and  oceans,  a  level  sur- 
face coincides  with  the  general  surface  of  the  earth.  Were 
the  earth  at  rest,  the  level  surface  of  lakes  and  oceans  would 
be  spherical ;  but,  on  account  of  the  centrifugal  force  aris- 
ing from  the  rotation  of  the  earth,  it  is  sensibly  an  ellip- 
soidal surface,  whose  axis  of  revolution  is  the  axis  of  the 
earth. 

Pressure  due  to  Weight. 

155.  If  an  incompressible  fluid  be  in  a  state  of  equili- 
brium, the  pressure  at  any  point  of  the  mass  arising  from 
the  weight  of  the  fluid,  is  proportional  to  the  depth  of  the 
point  below  the  free  surface. 

Take  an  infinitely  small  surface,  supposed  horizontal,  and 
conceive  it  to  be  the  base  of  a  vertical  prism  whose  altitude 
is  equal  to  its  distance  below  the  free  surface.  Conceive 
this  filament  to  be  divided  by  horizontal  planes  into  infi- 
nitely small,  or  elementary  prisms.  It  is  evident,  from  the 
principle  of  equal  })ressures,  that  the  pressure  upon  the 
lower  face  of  any  one  of  these  elementary  prisms  is  greater 
than  that  upon  its  upper  face,  by  the  weight  of  the  element, 
whilst  the  lateral  pressures  are  such  as  to  counteract  each 
other's  effects.  The  pressure  upon  the  lower  face  of  the 
first  prism,  counting  from  the  top,  is,  then,  just  equal  to  its 
weight ;  that  upon  the  lower  face  of  the  second  is  equal  to 
the  weight  of  the  first,  ]?lus  the  weight  of  the  second,  and 
so  on  to  the  bottom.  Hence,  the  pressure  upon  the  assumed 
surface  is  equal  to  the  weight  of  the  entire  column  of  fluid 
above  it.  Had  the  assumed  elementary  surface  been  oblique 
to  the  horizon,  or  perpendicular  to  it,  and  at  the  same  depth 
as  before,  the  pressure  upon  it  would  have  been  the  same, 
from  the  principle  of  equal  pressures.  We  have,  therefore, 
the  following  law : 

TJie  pressure  i/pon  any  elementary  portion  of  the  surface 
of  a  vessel  containing  a  heavy  fluid  is  equal  to  the  weight 
of  a  prism  of  the  fluid  whose  base  is  equal  to  that  surface, 


MECHANICS    OF   LIQUIDS.  239 

and  whose  altitude  is  equal  to  its  depth   below   the  free 
surface. 

Denoting  the  area  of  the  elementary  surface,  by  5,  its 
depth  below  the  free  surface,  by  z,  the  weight  of  a  unit  of 
the  volume  of  the  fluid,  by  w,  and  the  pressure,  by  p,  we 
shall  have, 

p  =  wzs ( 141.) 

We  have  seen  that  the  pressure  upon  any  element  of  a 
surface  is  normal  to  the  surface.  Denote 
the  angle  which  this  normal  makes  with 
the  vertical,  estimated  from  above,  down- 
wards, by  <p,  and  resolve  the  pressure  into 
two  components,  one  vertical  and  the 
other    horizontal,    denoting    the    vertical  —   135 

component  by  p',  we  shall  have, 

p'  —  iczscosp (142.) 

But  scoscp  is  equal  to  the  horizontal  projection  of  the 
elementary  surface  s,  or,  in  other  words,  it  is  equal  to  a 
horizontal  section  of  a  vertical  prism,  of  which  that  surface 
is  the  base.  Hence,  the  vertical  component  of  the  jtressitre 
on  any  element  of  the  surface  is  equal  to  the  iceight  of  a 
column  of  the  fluid,  whose  base  is  equal  to  the  horizontal 
p>rojection  of  the  element,  and  whose  altitude  is  equal  to 
the  distance  of  the  element  from  the  upper  surface  of  the 
fluid. 

The  distance  z  has  been  estimated  as  positive  from  the 
surface  of  the  fluid  downwards.  If  9  <  90°,  Ave  have  cos? 
positive  ;  hence,  p'  will  be  positive,  which  shows  that  the 
vertical  pressure  is  exerted  downwards.  If  <p  >  90°,  we 
have  cosp  negative  ;  hence,  p'  is  negative,  which  shows  that 
the  vertical  pressure  is  exerted  upwards  (see  Fig.  135). 

Suppose  the  interior  surface  of  a  vessel  containing  a  heavy 
fluid  to  be  divided  into  elementary  portions,  whose  areas 
are  denoted  by  s,  s\  s",  &c. ;  denote  the  distances  of  these 


24:0  MECHANICS. 

elements  below  the  upper  Surface,  by  z,  z',  z",  &c.     From 

the  principle  just  demonstrated,  the  pressures  upon  these 
surfaces  will  be  denoted  by  icsz,  ws'z,  ics"z",  &c,  and  the 
entire  pressure  upon  the  interior  of  the  vessel  will  be 
equal  to, 

w(sz  +  s'z  +  s"z"  +  &c.) ;     or,     to  x  2(sz). 

Let  Z  denote  the  depth  of  a  column  of  the  fluid,  whose 
base  is  equal  to  the  entire  surface  pressed,  and  whose  weight 
is  equal  to  the  entire  pressure,  then  will  this  pressure  be 
equal  to  w(s  +  s'  -+-  *"  +  &c)Z;  or,  wZ .  Is.  Equating 
these  values,  we  have, 

w.2(sz)  =wZ.l(s),         .'.     Z=^j    •    (1^3.) 

The  second  member  of  (143),  (Art.  51),  expresses  the 
distance  of  the  centre  of  gravity  of  the  surface  pressed, 
below  the  free  surface  of  the  fluid.     Hence, 

The  entire  pressure  of  a  heavy  fluid  upon  the  interior  of 
the  containing  vessel,  is  equal  to  the  weight  of  a  volume  of 
the  fluid,  u'/wse  base  is  equal  to  the  area  of  the  surface 
■pressed,  and  ichose  altitude  is  equal  to  the  distance  of  the 
centre  of  gravity  of  the  surface  from  the  free  surface  of  the 
fluid. 

EXAMPLES. 

1.  A  hollow  sphere  is  filled  with  a  liquid.  How  does  the 
entire  pressure,  on  the  interior  surface,  compare  with  the 
weight  of  the  liquid  ? 

SOLUTION. 

Denote  the  radius  of  the  interior  surface  of  the  sphere, 
by  r,  and  the  weight  of  a  unit  of  volume  of  the  liquid,  by 
w.  The  entire  surface  pressed  is  measured  by  4c?-2;  and, 
since  the  centre  of  gravity  of  the  surface  pressed  is  at  a 
distance  r  below  the  surface  of  the  liquid,  the  entire  pre* 


MKCHAXK.S    OF    LIQUIDS.  241 

sure   on    tne    interior    surface   will   be   measured  by   the 
expression, 

w  X  4^r5  x  r  —  4«tor*. 

But  the  weight  of  the  liquid  is  equal  to 

Hence,  the  entire  pressure  is  equal  to  three  times  the 
weight  of  the  liquid. 

2.  A  hollow  cylinder,  with  a  circular  base,  is  filled  with  a 
liquid.  How  does  the  pressure  on  the  interior  surface  com- 
pare with  the  weight  of  the  liquid? 

SOLUTION. 

Denote  the  radius  of  the  base  of  the  cylinder,  by  r,  and 
the  altitude,  by  h.  The  centre  of  gravity  of  the  lateral 
surface  is  at  a  distance  below  the  upper  surface  of  the  fluid 
equal  to  \h.  If  we  denote  the  weight  of  the  unit  of  volume 
of  the  liquid,  by  w,  we  shall  have,  for  the  entire  pressure  on 
the  interior  surface, 

whxr*  +  2wxr .  \h*  =  wirrh{r  -f-  h). 
But  the  weight  of  the  liquid  is  equal  to 
wtr^h. 

T  +  h 

Hence,  the  total  pressure  is  equal  to  times  the 

iceight  of  the  liquid. 

If  we  suppose  h  =  r,  the  pressure  will  be  twice  the 
weight. 

If  we  suppose  r  =  2h,  we  shall  have  the  pressure  equal 
to  \  of  the  weight. 

If  we  suppose  h  =;  2r,  the  pressure  will  be  equal  to  three 
times  the  weight,  and  so  on. 
11 


242  MECHANICS. 

In  all  cases,  the  total  pressure  will  exceed  tbe  weight  of 
the  liquid. 

3.  A  right  coue,  with  a  circular  base,  stands  on  its  base, 
and  is  tilled  with  a  liquid.  How  does  the  pressure  on  the 
internal  surface  compare  with  the  weight  of  the  liquid  ? 

solution. 

Denote  the  radius  of  the  base,  by  r,  and  the  altitude,  by 
A,  then  will  the  slant  height  be  equal  to 

^/hF^?. 

The  centre  of  gravity  of  the  lateral  surface,  below  the 
upper  surface  of  the  liquid  is  equal  to  §A.  If  we  denote 
the  weight  of  a  unit  of  volume  of  the  liquid,  by  w,  we  shall 
have,  for  the  total  pressure  on  the  interior  surface, 


wvr'h  +  %w*rhi/h*  +  r*  =  w«rh{r  +  f -/A2  +  r3). 
But  the  weight  of  the  liquid  is  equal  to 
%ioirr*h  =  w*rh  x  £r. 

3r  +  2v/ArT~ra 


Hence,  the  total  pressure  is  equal  to 
times  the  iceight. 

4.  Required  the  relation  between  the  pressure  and  the 
weight  in  the  preceding  case,  when  the  cone  stands  on  its 
vertex. 

SOLUTION. 

The  total  pressure  is  equal  to 


^wrrhyh2  4-  r° ; 


■\Zha  -f  r* 
and,  consequently,  the  pressure  is  equal  to  — times 


the  weight  of  the  liquid 


MECHANICS    OF    LIQUIDS.  243 

5.  What  is  the  pressure  on  the  lateral  faces  of  a  cubical 
vessel  filled  with  water,  the  edges  of  the  cube  being  4  feet, 
and  the  weight  of  the  water  62^  lbs.  per  cubic  foot  ? 

A?is.  8000  lbs. 

6.  A  cylindrical  vessel  is  filled  with  water.  The  height 
of  the  vessel  is  4  feet,  and  the  radius  of  the  base  6  feet. 
What  is  the  pressure  on  the  lateral  surface  ? 

Ans.  18850  lbs.,  nearly. 

Centre  of  Pressure  on  a  Plane  Surface. 

156.     Let  ABCD  represent  a  plane,  pressed  by  a  fluid 
on  its  upper  surface,  AB  its  intersec- 
tion with  the  free  surface  of  the  fluid,         ,  j± 

G  its  centre  of  gravity,  0  the  centre       y- ^^^^^ 

of  pressure,   and  s   the   area  of  any  </   jj/    7 

element   of   the    surface    at    S.     De-  /W)G  / 

note  the  inclination  of  the  plane  to  4*  / 

the  level  surface,  by  a,  the  perpendic-  ^"^ 

ular  distances  from  0  to  AB,  by  x,  rig.  136. 

from  G  to  AB,  by  p,  and  from  S  to 

AB,  by  r.  Denote,  also,  the  entire  area  A  C,  by  A,  and 
the  weight  of  a  unit  of  volume  of  the  fluid,  by  w.  The 
perpendicular  distance  from  G  to  the  free  surface  of  the 
fluid,  will  be  equal  to  p  sina,  and  that  of  any  element  of  the 
surface,  will  be  r  sina. 

From  the  preceding  article,  it  follows  that  the  entire 
pressure  exerted  is  equal  to  wAp  sina,  and  its  moment,  with 
respect  to  AB  as  an  axis  of  moments,  is  equal  to 

wAp  sina  x  x. 

The  elementary  pressure  on  s  is,  in  like  manner,  equal  to 
wsrsina,  and  its  moment,  with  respect  to  AB,  is  wsr2sina, 
and  the  sum  of  all  the  elementary  moments  is  equal  to 

w  sina  ^(sr3). 


244  MECHANICS. 

But  the  resultant  moment  is  equal  to  the  algebraic  sum 
of  the  elementary  moments.     Hence, 

wAp  sina  x  x  —  w  sina  2(sr2)  ; 

and,  by  reduction, 

The  numerator  is  the  moment  of  inertia  of  the  plane 
ABCD,  with  respect  to  AB,  and  the  denominator  is  the 
moment  of  the  area  with  respect  to  the  same  line.  Hence, 
the  distance  from  the  centre  of  pressure  to  the  intersection 
of  the  plane  with  the  free  surface,  is  equal  to  the  moment 
of  inertia  of  the  plane,  divided  by  the  moment  of  the 
plane. 

If  we  take  the  straight  line  AD,  perpendicular  to  A B,  as 
an  axis  of  moments,  denoting  the  distance  of  0  from  it,  by 
y,  and  of  s  from  it,  by  /,  we  shall,  in  a  similar  manner,  have, 

wApsiwy  —  wsina2  (srZ); 

and,  by  reduction, 

y=4T (140^ 

The  values  of  x  and  y  make  known  the  position  of  the 
centre  of  pressure. 

EXAMPLES. 

1.  What  is  the  position  of  the  centre  of  pressure  on  a 
rectangular  flood-gate,  the  upper  line  of  the  gate  coinciding 
with  the  surface  of  the  water? 

SOLUTION. 

It  is  obvious  that  it  will  be  somewhere  on  the  line  joining 
the  middle  points  of  the  upper  and  lower  edges  of  the  gate. 


MECHANICS    OF    LIQUIDS.  245 

Denote  its  distance  from  the  upper  edge,  by  z,  the  depth  of 
the  gate-,  by  2/,  and  its  mass,  by  31.  The  distance  of  the 
centre  of  gravity  from  the  upper  edge  will  be  equal  to  I. 

From  Example  1  (Art.  132),  replacing  d  by  I,  and 
reducing,  we  have,  for  the  moment  of  inertia  of  the 
rectangle, 

jw(|  +  r)=  Mi  p. 

But  the  moment  of  the  rectangle  is  equal  to, 

Ml; 

hence,  by  division,  we  have, 

z  =  £  =  1(91). 

That  is,  the  centre  of  pressure  is  at  two-thirds  of  the 
distance  from  the  upper  to  the  lower  edge  of  the  gate. 

2.  Let  it  be  required  to  find  the  pressure  on  a  submerged 
rectangular  flood-gate  ABCD,  the  plane  of 
the  gate  being  vertical.     Also,  the   distance  E    g   I 

of  the  centre  of  pressure  below  the  surface 
of  the  water. 


SOLUTION. 


C 

— IB 

C 

jC" 


3  C 

Fig.  137. 


Let  EF  be  the  intersection  of  the  plane 
with  the  surface  of  the  water,  and  suppose 
the  rectangle  AC  to  be  prolonged  till  it 
reaches  EF.  Let  C,  C\  and  C'\  be  the  centres  of  pressure 
of  the  rectangles  EC,  EB,  and  A  C  respectively.  Denote 
the  distance  (jC'\  by  2,  the  distance  ED,  by  a,  and  the 
distance  EA,  by  a'.  Denote  the  breadth  of  the  gate,  by  b, 
and  the  weight,  a  unit  of  volume  of  the  water,  by  to. 

The  pressure  on  EC  will  be  equal  to  Ufbw,  and  the  pres- 
sure on  EB  will  be  equal  to  \a'2bw ;  hence,  the  pressure  on 
AC  will  be  equal  to 

^bwijtf  —  a'*)  ; 

which  is  the  pressure  required. 


246  MECHANICS. 

From  the  principle  of  moments,  the  moment  of  the  pres- 
sure on  A  C,  is  equal  to  the  moment  of  the  pressure  on  EC, 
minus  the  moment  of  the  pressure  on  EB.  Hence,  from 
the  last  problem, 

^bw(ai  —  a'2)  x  z  =  ^bica?  x  § a  —  \bwan  x  fa', 


ir 


which  is    the   required  distance   from  the   surface   of  the 
water. 

3.  Let  it  be  required  to  find  the  pressure  on  a  rectangular 
flood-gate,  when  both  sides  are  pressed, 
the  water  being  at  different  levels  on          wpgmilg^- 
the  two  sides.   Also,  to  find  the  centre 
of  pressure.  fa 

solution.  L 

Denote  the  depth  of  water  on  one  Fi    m 

side  by  «,  and  on  the  other  side,  by 
a\  the  other  elements  being  the  same  as  before. 
The  total  pressure  will,  as  before,  be  equal  to, 

ibwia'*  -  a"). 

Estimating  z  from  C  upwards, 


Arts. 


4.  A  sluice-gate,  10  feet  square,  is  placed  vertically,  its 
upper  edge  coinciding  with  the  surface  of  the  water  What 
is  the  pressure  on  the  upper  and  lower  halves  of  the  gate, 
respectively,  the  weight  of  a  cubic  foot  of  water  being 
taken  equal  to  621  lbs.?       A?is.  ^812.5  lbs.,  and  23437.5  lbs. 

5.  What  must  be  the  thickness  of  a  rectangular  dam  of 
granite,  that  it  may  neither  rotate  about  its  outer  angular 


MECHANICS    OF    LIQUIDS.  247 

point  nor  slide  along  its  base,  the  weight  of  a  cubic  foot  of 
granite  being  160  lbs.,  and  the  coefficient  of  friction  between 
it  and  the  soil  being  .6  ? 

SOLUTION. 

First,  to  find  the  thickness  necessary  to  prevent  rotation 
outwards.  Denote  the  height  of  the  wall,  by  hy  and  sup- 
pose the  water  to  extend  from  the  bottom  to  the  top.  De- 
note the  thickness,  by  t,  and  the  length  of  the  wall,  or  dam, 
by  I.     The  weight  of  the  wall  in  pounds,  will  be  equal  to 

Iht  x  160  ; 

and  this  being  exerted  through  its  centre  of  gravity,  the 
moment  of  the  weight  with  respect  to  the  outer  edge,  as  an 
axis,  will  be  equal  to 

\flh  X  160  =  80lht\ 


The  pressure  of  the  water  against  the  inner  face,  in 
pounds,  is  equal  to 

l/A2   X62.5  =  lh?  X   31.25. 

This  pressure  is  applied  at  the  centre  of  pressure,  which 
is  (Example  I)  at  a  distance  from  the  bottom  of  the  wall 
equal  to  ^h ;  hence,  its  moment  with  respect  to  the  outer 
edge  of  the  wall,  is  equal  to 

W   X  10.4166. 

The  pressure  of  the  water  tends  to  produce  rotation  out- 
wards, and  the  weight  of  the  wall  acts  to  prevent  this  rota- 
tion. In  order  that  these  forces  may  be  in  equilibrium, 
their  moments  must  be  equal ;  or 

80lhf  -  lh%  X   10.4166. 


'2±S  MECHANICS. 

Whence,  we  find, 


t  =  h  y/.  1302  =  .36   X  h. 

■ 

Next,  to  find  the  thickness  necessary  to  prevent  sliding 
alone:  the  base.  The  entire  force  of  friction  due  to  the 
weight  of  the  wall,  is  equal  to 

IQOlht  X  .6  —  96lht; 

and  in  order  that  the  wall  may  not  slide,  this  must  be  equal 
to  the  pressure  exerted  horizontally  against  the  wall.    Hence, 

96lht  =  3l.2olh\ 

Whence,  we  find, 

t  =  .325A. 

If  the  wall  is  made  thick  enough  to  prevent  rotation,  it 
will  be  secure  against  sliding. 

6.  What  must  be  the  thickness  of  a  rectangular  dam 
15  feet  high,  the  weight  of  the  material  being  140  lbs.  to 
the  cubic  foot,  that,  when  the  water  rises  to  the  top,  the 
structure  may  be  just  on  the  point  of  overturning  ? 

Ans.  5.7  ft. 

7.  The  staves  of  a  cylindrical  cistern  filled  with  water,  are 
held  together  by  a  single  hoop.  Where  must  the  hoop  be 
situated  ? 

Ans  At  a  distance  from  the  bottom  equal  to  one-third  of 
the  height  of  the  cistern. 

8.  Required  the  pressure  of  the  sea  on  the  cork  of  an 
empty  bottle,  when  sunk  to  the  depth  of  000  feet,  the 
diameter  of  the  cork  being  J  of  :m  inch,  and  a  cubic  foot  of 

sea  water  being  estimated  to  weigh  64  lbs.?     Ans.  134  lbs. 


A0 


MECHANICS    OF    LIQUIDS.  249 

Buoyant  Effort  of  Fluids. 

157.  Let  A  represent  any  solid  body  suspended  in  a 
heavy  fluid.  Conceive  this  solid  to  be  divided 
into  vertical  prisms,  whose  horizontal  sections  are 
infinitely  small.  Any  one  of  these  prisms  will  be 
pressed  downward  by  a  force  equal  to  the  weight 
of  a  column  of  fluid,  whose  base  (Art.  155)  is  Fi  139" 
equal  to  the  horizontal  section  of  the  filament, 
and  whose  altitude  is  the  distance  of  its  upper  surface  from 
the  surface  of  the  fluid ;  it  will  be  pressed  upward  by  a 
force  equal  to  the  weight  of  a  column  of  fluid  having  the 
same  base  and  an  altitude  equal  to  the  distance  of  the  lower 
base  of  the  filament  from  the  surface  of  the  fluid.  The  re- 
sultant of  these  two  pressures  is  a  force  exerted  vertically 
upwards,  and  is  equal  to  the  weight  of  a  column  of  fluid, 
equal  in  bulk  to  that  of  the  filament  and  having  its  point 
of  application  at  the  centre  of  gravity  of  the  volume  of  the 
filament.  This  being  true  for  each  filament  of  the  body, 
and  the  lateral  pressures  being  such  as  to  destroy  each 
other's  effects,  it  follows,  that  the  resultant  of  all  the  pres- 
sures upon  the  body  will  be  a  vertical  force  exerted  upwards, 
whose  intensity  is  equal  to  the  weight  of  a  portion  of  the 
fluid,  whose  volume  is  equal  to  that  of  the  solid,  and  the 
point  of  application  of  which  is  the  centre  of  gravity  of  the 
volume  of  the  displaced  fluid.  This  upward  pressure  is  call- 
ed the  buoyant  effort  of  the  fluid,  and  its  point  of  application 
is  called  the  centre  of  buoyancy.  The  line  of  direction  of 
the  buoyant  effort,  in  any  position  of  the  body,  is  called  a 
line  of  support.  That  line  of  support  which  passes  through 
the  centre  of  gravity  of  a  body,  is  called  the  line  of  rest. 

Floating  Bodies. 

158.     A  body  wholly  or  partially  immersed  in  a  heavy 
fluid,  is  urged  downwards  by  its  weight  applied  at  its  cen- 
tre of  gravity,  and  upwards,  by  the  buoyant  effort  of  the 
fluid  applied  at  the  centre  of  buoyancy. 
11* 


'250 


MKCHANIC8. 


sr 


The  body  can  only  be  in  equilibrium  when  the  line  through 
the  centre  of  gravity  of  the  body,  and  the  centre  of  buoy- 
ancy, is  vertical  ;  in  other  words,  when  the  line  of  rest  is  ver- 
tical. When  the  weight  of  the  body  exceeds  the  buoyant 
effort,  the  body  will  sink  to  the  bottom  ;  when  they  are 
just  equal,  it  will  remain  in  equilibrium,  wherever  placed  in 
the  fluid.  When  the  buoyant  effort  is  greater  than  the 
weight, it  will  rise  to  the  surface,  and  after  a  few  oscillations, 
will  come  to  a  state  of  rest,  in  such  a  position,  that  the 
weight  of  the  displaced  fluid  is  equal  to  that  of  the  body, 
when  it  is  said  to  float.  The  upper  surface  of  the  fluid  is 
then  called  the  i^lane  of  floatation,  and  its  intersection  with 
the  surface  of  the  body,  the  line  of  floatation. 

If  a  floating  body  be  slightly  disturbed  from  its  position 
of  equilibrium,  the  centres  of  grav- 
ity and  buoyancy  will  no  longer 
be  in  the  same  vertical  line.  Let 
DJE  represent  the  plane  of  floata- 
tion, G  the  centre  of  gravity  of  the 
body  (Fig.  141),  Gil  its  line  of  rest, 
and  C  the  centre  of  buoyancy  in 
the  disturbed  position  of  the 
body. 

If  the  line  of  support  CB,  in- 
tersects  the  line  of  rest  in  31, 
above  6r,  as  in  Fig.  141,  the  buoy- 
ant effort  and  the  weight  will  conspire  to  restore  the  body 
to  its  position  of  equilibrium  ;  in  this  case,  the  equilibrium 
must  be  stable. 

If  the  point  M  falls  below  G, 
as  in  Fig.  142,  the  buoyant  ef- 
fort and  the  weight  will  conspire 
to  overturn  the  body ;  in  this 
case,  the  body  must,  before  be- 
ing disturbed,  have  been  in  a 
state  of  unstable  equilibrium. 

If  the  centre  of   buoyancy  and  centre  of   gravity  are 


Fig.  140. 


MECHANICS    OF   LIQUIDS.  251 

always  on  the  same  vertical,  the  point 
M  will  coincide  with  G  (Fig.  143), 
and  the  body  will  be  in  a  state  of 
indifferent  equilibrium.  The  limiting 
position  of  the  point  31,  or  of  the 
intersection  of  the  lines  of  rest  and  ™    ,,0 

Jng.  143. 

of  support,  obtained  by  disturbing  the 

floating  body  through  an  infinitely  small  angle,  is  called  the 

metacentre  of  the  body.     Hence, 

If  the  metacentre  is  above  the  centre  of  gravity  of  the 
body,  it  will  be  in  a  state  of  stable  equilibrium,  the  line  of 
rest  being  vertical;  if  it  is  below  the  centre  of  gravity,  the 
body  will  be  in  unstable  equilibrium  ;  if  the  two  points 
coincide,  the  body  will  be  in  indifferent  equilibrium. 

The  stability  of  the  floating  body  will  be  the  greater,  as 
the  metacentre  is  higher  above  the  centre  of  gravity.  This 
condition  is  practically  fulfilled  in  loading  ships,  or  other 
floating  bodies,  by  stowing  the  heavier  objects  nearest  the 
bottom  of  the  vessel. 

Specific  Gravity. 

159.  The  specific  gravity  of  a  body  is  its  relative  weight ; 
that  is,  it  is  the  number  of  times  the  body  is  heavier  than 
an  equivalent  volume  of  some  other  body  taken  as  a 
standard. 

The  numerical  value  of  the  specific  gravity  of  any  body, 
is  the  quotient  obtained  by  dividing  the  weight  of  any 
volume  of  the  body  by  that  of  an  equivalent  volume  of  the 
standard. 

For  solids  and  liquids,  water  is  generally  taken  as  the 
standard,  and,  since  this  liquid  is  of  different  densities  at 
different  temperatures,  it  becomes  necessary  to  assume  also 
a  standard  temperature.  Most  writers  have  taken  60° 
Fahrenheit  as  this  standard.  Some,  however,  have  taken 
3 8° 75  Fah.,  for  the  reason  that  experiment  has  shown  that 
water  has  its  maximum  density  at  this  temperature.  We 
shall   adopt   the   latter    standard,  remarking   that    specific 


252  MECHANICS. 

gravities,  determined  at  any  temperature,  may  be  readily 
reduced  to  what  they  would  have  been  had  they  been  deter- 
mined at  any  other  temperature. 

The  densities  of  pure  water  at  different  temperatures  has 
been  determined  with  great  accuracy  by  experiment,  and 
the  results  arranged  in  tables,  the  density  at  38°75  being 
taken  as  1. 

Since  the  specific  gravity  of  a  body  increases  as  the 
density  of  the  standard  diminishes,  it  will  be  a  little  less 
when  referred  to  water  at  38°75  than  at  any  other  tempe- 
rature. 

Let  d  and  d'  denote  the  densities  of  water  at  any  two 
temperatures  t  and  t' ;  let  s  and  s'  denote  the  specific 
gravities  of  the  same  body,  referred  to  water  at  these 
temperatures ;  then, 

s'd' 
s  :  s'  :  :  d'  :  d,         .-.     s  =  —     .     (  146.) 

a 

This  formula  is  applicable  in  any  case  where  it  is  necessary 
to  reduce  the  specific  gravity  taken  at  the  temperature  t' 
to  what  it  would  have  been  if  taken  at  the  temperature  t. 
If  t  =  38°75,  we  have  d  =  1,  and  the  formula  becomes, 

s  =  s'd' (147.) 

Hence,  to  reduce  the  specific  gravity  taken  at  the  tem- 
perature t\  to  the  standard  temperature,  multiply  it  by 
the  tabular  density  of  water  at  the  temperature  t'. 

The  specific  gravity  should  also  be  corrected  for  expan- 
sion. This  correction  is  made  in  a  manner  entirely  similar 
to  the  last.  Denote  the  volumes  of  the  same  body  at  the 
temperatures/  and  t',  by  v  and  v',  and  the  apparent  specific 
gravities,  after  the  last  correction,  by  S  and  N',  then, 

8  :  S'  :  :  v'  :  v,         .\     S  =  —  (H8.) 

7  v 


MECHANICS    OF    LIQUIDS. 


253 


If  t  is  the  standard  temperature,  and  v  the  unit  of  volume 
we  have, 

S  =  &  XV'      .     .     .     .     ( 149.) 


In  what  follows,  we  shall  suppose  that  the  specific  gravi- 
ties are  taken  at  the  standard  temperature,  in  which  case 
no  correction  will  be  necessary. 

Gases  are  generally  referred  to  atmospheric  air  as  a 
standard,  but,  as  air  may  be  readily  referred  to  water  as  a 
standard,  we  shall,  for  the  purpose  of  simplification,  suppose 
that  the  standard  for  all  bodies  is  distilled  water  at  38°75 
Fahrenheit. 

Hydrostatic  Balance. 

160.     This  balance  is  similar  to 
that  described  in  Article  81,  ex-  > 

cept  the  scale-pans  have  hooks  at- 
tached to  their  lower  surfaces  for 
the  purpose  of  suspending  bodies. 
The  suspension  is  effected  by  a 
fine  platinum  wire,  or  by  some  — 
other  material  not  acted  upon  by 
the  liquids  employed. 


n 


Fig.  144 


To  determine  the  Specific  Gravity  of  an  Insoluble  Body. 

161.  Attach  the  suspending  wire  to  the  first  scale-pan, 
and  after  allowing  it  to  sink  in  a  vessel  of  water  to  a  certain 
depth,  counterpoise  it  by  an  equal  weight,  attached  to  the 
hook  of  the  second  scale-pan.  Place  the  body  in  the  first 
scale-pan,  and  counterpoise  it  by  weights  in  the  second  pan. 
These  weights  will  give  the  weight  of  the  body  in  air. 
Xext,  attach  the  body  to  the  suspending  wire,  and  immerse 
it  in  the  water.  The  buoyant  effort  of  the  Mater  will  be 
equal  to  the  weight  of  a  volume  of  water  equivalent  to  that 
of  the  body  (Art.  157)  ;  hence,  the  second  pan  will  descend. 
Restore  the  equilibrium  by- weights  placed  iti  the  first  pan. 
These  weights  will  give  the  weight  of  the  displaced  water. 


254  MECHANICS. 

Divide  the  weight  of  the  body  in  air  by  the  weight  just 
found,  and  the  quotient  will  be  the  specific  gravity  sought. 
If  the  body  will  not  sink  in  water,  determine  its  weight  in 
air  as  before ;  then  attach  to  it  a  body  so  heavy,  that  the 
combination  will  sink  ;  find,  as  before,  the  loss  of  weight  of 
the  combination,  and  also  the  loss  of  weight  of  the  heavier 
body ;  take  the  latter  from  the  former,  and  the  difference 
will  be  the  loss  of  weight  of  the  lighter  body ;  divide  its 
weight  in  air  by  this  weight,  and  the  quotient  will  be  the 
specific  gravity  sought. 

If  great  accuracy  is  required,  account  must  be  taken  of 
the  buoyant  effort  of  the  air,  which,  when  the  body  is  very 
light,  and  of  considerable  dimensions,  will  render  the  appa- 
rent weight  less  than  the  true  weight,  or  the  weight  in 
vacuum.  Since  the  weights  used  in  counterpoising  are 
always  very  dense,  and  of  small  dimensions,  the  buoyant 
effort  of  the  air  upon  them  may  always  be  neglected. 

To^determine  the  true  weight  of  a  body  in  vacuum  :  let 
to  denote  its  weight  in  air,  w'  its  weight  in  water,  and  IF  its 
weight  in  vacuum  ;  then  will  W  —  ?c,  and  IV —  w',  denote 
its  loss  of  weight  in  air  and  water ;  denote  the  specific 
gravity  of  air  referred  to  water,  by  s.  Since  the  losses  of 
weight  in  air  and  water  arc  proportional  to  their  specific 
gravities,  we  have, 

W  —  w  :  W  —  w'  :  :  s  :  1  ;     or,    W  —  to  =  *  W  —  sio\ 

1  —  s 
This  weight  should  be  used,  instead  of  the  weight  in  air. 

To  determine  the  Specific  Gravity  of  Liquids. 

16*2.  First  Method. — Take  a  vial  with  a  narrow  neck, 
and  weigh  it ;  fill  it  with  the  liquid,  and  weigh  again ; 
empty  out  the  liquid,  and  fill  with  water,  and  weigh  again  ; 
deduct  from  the  last  two  weights,  respectively,  the  weight 
of  the   vial;  these  results  will  give  the  weights  of- equal 


MECHANICS    OF    LIQUID?.  255 

volumes  of  the  liquid  and  of  water.  Divide  the  former  by 
the  latter,  and  the  quotient  will  be  the  specific  gravity 
sought. 

Second  Method. — Take  a  heavy  body,  that  will  sink  both 
in  the  liquid  and  in  water,  and  which  will  not  be  acted  upon 
by  either  ;  determine  its  loss  of  weight,  as  already  explained, 
first  in  the  liquid,  then  in  water;  divide  the  former  by  the 
latter,  and  the  quotient  will  be  the  specific  gravity  sought. 
The  reason  is  evident. 

Third  Method. — Let  AB  and  CD  represent  two 
graduated  glass  tubes  of  half  an  inch  in 
diameter,  open  at  both  ends.  Let  their 
upper  ends  communicate  with  the  receiver 
of  an  air-pump,  and  their  lower  ends  dip 
into  two  cisterns,  one  containing  distilled 
water,  and  the  other  the  liquid  whose 
specific  gravity  is  to  be  determined.  Let 
the  air  be  partially  exhausted  from  the 
receiver  by  means  of  an  air-pump ;  the  liquids  will  rise  in 
the  tubes,  but  to  different  heights,  these  being  inversely  as 
the  specific  gravities  of  the  liquids.  If  we  divide  theheigl  t 
of  the  column  of  water  by  that  of  the  other  liquid,  the 
quotient  will  be  the  specific  gravity  sought.  By  creating 
different  degrees  of  rarefaction,  the  columns  will  rise  to 
different  heights,  but  their  ratios  ought  to  be  the  same.  We 
are  thus  enabled  to  make  a  series  of  observations,  each  cor- 
responding to  a  different  degree  of  rarefaction,  from  which 
a  more  accurate  result  can  be  had  than  from  a  single  obser- 
vation. 

To  determine  the  Specific  Gravity  of  a  Soluble  Body. 

163.  Find  its  specific  gravity  by  the  method  already 
given,  with  respect  to  some  liquid  in  which  it  is  not  soluble, 
and  find  also  the  specific  gravity  of  this  liquid  referred  to 
water;  take  the  product  of  these  specific  gravities,  and  it 
will  be  the  specific  gravity  sought.  For,  if  the  body  is  m 
times  heavier  than  an  equivalent  volume  of  the  liquid  used, 


256 


MECHANICS. 


and  this  is  n  times  heavier  than  an  equivalent  volume  of 
water,  it  follows  that  the  body  is  mn  times  heavier  than  its 
volume  of  water,  whence  the  rule. 

The  auxiliary  liquid,  in  some  cases,  might  be  a  saturated  solu- 
tion of  the  given  body  in  water  ;.  the  rule  remains  unchanged. 

To  determine  the  Specific  Gravity  of  the  Air. 
164.  Take  a  hollow  globe,  fitted  with  a  stop-cock,  to 
shut  off  communication  with  the  external  air,  and,  by  means 
of  the  air-pump  or  condensing  syringe,  pump  in  as  much  air 
as  is  convenient,  close  the  stop-cock,  and  weigh  the  globe 
thus  filled.  Provide  a  glass  tube,  graduated  so  as  to  show 
cubic  inches  and  decimals  of  a  cubic 
inch,  and,  having  tilled  it  with  mer- 
cury, invert  it  over  a  mercury  bath. 
Open  the  stopcock,  and  allow  the  com- 
pressed air  to  escape  into  the  inverted 
tube,  taking  care  to  bring  the  tube 
into  such  a  position  that  the  mercury 
without  the  tube  is  at  the  same  level 
as  within.     The  reading  on  the  tube 

will  give  the  volume  of  the  escaped  air.  Weigh  the  globe 
again,  and  subtract  the  weight  thus  found  from  the  first 
weight ;  this  difference  will  indicate  the  weight  of  the 
escaped  air.  Having  reduced  the  measured  volume  of  air 
to  what  it  would  have  occupied  at  a  standard  temperature 
and  barometric  pressure,  by  means  of  rules  yet  to  be 
deduced,  compute  the  weight  of  an  equivalent  volume  of 
water;  divide  the  weight  of  the  corrected  volume  of  air  by 
that  of  an  equivalent  volume  of  distilled  water,  and  the 
quotient  will  be  the  specific  gravity  sought. 

To  determine  the  Specific  Gravity  of  a  Gas. 
IG5.  Take  a  glass  globe  of  suitable  dimensions,  fitted 
with  a  stop-cook  for  shutting  off  communication  with  the 
atmosphere.  Fill  the  globe  with  air,  and  determine  the 
weight  of  the  globe  thus  filled  referred  to  a  vacuum,  as 
already  explained.     From  the  known  volume  of  the  globe 


Fig.  146. 


MECHANICS    OF    LIQUIDS.  257 

and  the  specific  gravity  of  air,  the  weight  of  the  contained 
air  can  be  computed ;  subtract  this  from  the  previous 
weight,  and  we  shall  have  the  true  weight  of  the  globe 
alone;  determine  in  succession  the  weights  of  the  globe 
filled  with  water  and  with  the  gas  in  vacuum,  and  from  each 
subtract  the  weight  of  the  globe ;  divide  the  latter  result  by 
the  former;  the  quotient  will  be  the  specific  gravity  required. 

Hydrometers. 

166.  A  hydrometer  is  a  floating  body,  used  for  the  pur- 
pose of  determining  specific  gravities.  Its  construction  de- 
pends upon  the  principle  of  floatation.  Hydrometers  are 
of  two  kinds.  1.  Those  in  which  the  submerged  volume  is 
constant.  2.  Those  in  which  the  weight  of  the  instrument 
remains  constant. 

Nicholson's  Hydrometer. 

167.  This  instrument  consists  of  a  hollow  brass  cylinder 
A,  at  the  lower  extremity  of  which  is  fastened 

a  basket  B,  and  at  the  upper  extremity  a  wire, 
bearing  a  scale-pan  C.  At  the  bottom  of  the 
basket  is  a  ball  of  glass  E,  containing  mer- 
cury, the  object  of  which  is,  to  cause  the  in- 
strument to  float  in  an  upright  position.  By 
means  of  this  ballast,  the  instrument  is  ad- 
justed so  that  a  weight  of  500  grains,  placed 
in  the  pan  C\  will  sink  it  in  distilled  water  to 
a  notch  2>,  filed  in  the  neck. 

To  determine  the  specific  gravity  of  a  solid  Fig.  147. 

which  weighs  less  than  500  grains.  Place  the 
body  in  the  pan  C,  and  add  weights  till  the  instrument 
sinks,  in  distilled  water,  to  the  notch  J).  The  added 
weights,  substracted  from  500  grains,  will  give  the  weight 
of  the  body  in  air.  Place  the  body  in  the  basket  JB,  which 
generally  has  a  reticulated  cover,  to  prevent  the  body  from 
floating  away,  and  add  other  weights  to  the  pan,  until  the 
instrument  again  sinks  to  the  notch  D.  The  weights  last 
added  give  the  weight  of  the  water  displaced  by  the  body. 


258  MECHANICS. 

Divide  the  first  of  these  -weights  by  the  second,  and  tne 
quotient  will  be  the  specific  gravity  required. 

To  find  the  specific  gravity  of  a  liquid.  Having  carefully 
weighed  the  instrument,  place  it  in  the  liquid,  and  add 
weights  to  the  scale-pan  till  it  sinks  to  D.  The  weight  of 
the  instrument,  plus  the  sum  of  the  weights  added,  will  be 
the  weight  of  the  liquid  displaced  by  the  instrument.  Next, 
place  the  instrument  in  distilled  water,  and  add  weights  till 
it  sinks  to  I).  The  weight  of  the  instrument,  plus  the  added 
weights,  gives  the  weight  of  the  displaced  water.  Divide 
the  first  result  by  the  second,  and  the  quotient  will  be  the 
specific  gravity  required.     The  reason  for  this  rule  is  evident. 

A  modification  of  this  instrument,  in  which  the  basket  B, 
is  omitted,  is  sometimes  constructed  for  determining  specific 
gravities  of  liquids  only.  This  kind  of  hydrometer  is 
generally  made  of  glass,  that  it  may  not  be  acted  upon 
chemically,  by  the  liquids  into  which  it  is  plunged.  The 
hydrometer  just  described,  is  generally  known  as  Fahren 
heiVs  hydrometer,  or  Fahrenheit's  areometer. 

Scale  Areometer. 
168.     The  scale  areometer  is  a  hydrometer  whose  weight 
remains  constant ;  the  specific  gravity  of  a  liquid  is  made 
known  by  the  depth  to  which  it  sinks  in  it.     The 
instrument  consists  of  a  hollow  glass  cylinder  A,  n 

with  a  stem  (7,  of  uniform  diameter.  At  the 
bottom  of  the  cylinder  is  a  bulb  B,  containing 
mercury,  to  make  the  instrument  float  upright. 
By  introducing  a  suitable  quantity  of  mercury, 
the  instrument  may  be  adjusted  so  as  to  float  at 
any  desired  point  of  the  stem.  When  it  is  de- 
signed to  determine  the  specific  gravities  of  liquids, 
both  heavier   and    lighter  than  water,  it  is  bal-  Bo 

lasted  so  that  in  distilled  water,  it  will  sink  to  the  n..  14-. 

middle  of  the  stem.     This  point  is  marked  on  the 
stem  with  a  file,  and  since  the  specific  gravity  of  water  is  1, 
it  is  numbered   1   on  the  scale.     A  liquid  is  then  formed  by 
dissolving  common  salt  in  water  whose  specific  gravity  is 


MECHANICS    OF    LIQUIDS.  259 

1.1,  and  the  instrument  is  allowed  to  float  freely  in  it;  the 
point  E,  to  which  it  then  sinks,  is  marked  on  the  stem,  and 
the  intermediate  part  of  the  scale,  HE,  is  divided  into  10 
equal  parts,  and  the  graduation  continued  above  and  below 
throughout  the  stem.  The  scale  thus  constructed  is  marked 
on  a  piece  of  paper  placed  within  the  hollow  stem.  To  use  this 
hydrometer,  we  have  simply  to  put  it  into  the  liquid  and 
allow  it  to  come  to  rest;  the  division  of  the  scale  which  cor- 
responds to  the  surface  of  floatation,  makes  known  the  spe- 
cific gravity  of  the  liquid.  The  hypothesis  on  which  this 
instrument  is  graduated,  is,  that  the  increments  of  specific 
gravity  are  proportional  to  the  increments  of  the  submerged 
portion  of  the  stem.  This  hypothesis  is  only  approximately 
true,  but  it  approaches  more  nearly  to  the  truth  as  the  dia- 
meter of  the  stem  diminishes. 

When  it  is  only  desired  to  use  the  instrument  for  liquids 
heavier  than  water,  the  instrument  is  ballasted  so  that  the 
division  1  shall  come  near  the  top  of  the  stem.  If  it  is  to 
be  used  for  liquids  lighter  than  water,  it  is  ballasted  so  that 
the  division  1  shall  fall  near  the  bottom  of  the  stem.  In 
this  case  we  determine  the  point  0.9  by  using  a  mixture  of 
alcohol  and  water,  the  principle  of  graduation  being  the  same 
as  in  the  first  instance. 

Volumeter. 
169.     The  volumeter  is  a  modification  of  the  scale  areo- 
meter, differing  from  it  only  in  the  method  of  graduation. 
The  graduation  is  effected  as  follows :   The  instru- 
ment is  placed  in  distilled  water,  and  allowed  to 
come  to  a  state  of  rest,  and  the  point  on  the  stem 
where  the  surface  cuts  it,  is  marked  with  a  file. 
The  submerged  volume  is  then  accurately  deter- 
mined, and  the  stem  is  graduated  in  such  a  man- 
ner that  each  division  indicates  a  volume  equal  to 
a  hundredth  part  of  the  volume  originally  sub- 
merged.    The  divisions  are  then  numbered  from 
the  first  mark  in  both  directions,  as  indicated  in 
the  figure.     To  use  the  instrument,  place  it  in  the     *lg' 149' 
iquid,  and  note  the  division  to  which  it  sinks ; 


C 


A 


BV 


260  MECHANICS. 

divide  100  by  the  number  indicated,  and  the  quotient  will 
be  the  specific  gravity  sought.  The  principle  employed  is, 
that  the  specific  gravities  of  liquids  are  inversely  as  the  vol- 
umes of  equal  weights.  Suppose  that  the  instrument  indi- 
cates x  parts ;  then  the  weight  of  the  instrument  displaces 
x  parts  of  the  liquid,  whilst  it  displaces  100  parts  of 
water.  Denoting  the  specific  gravity  of  the  liquid  by  S,  and 
that  of  water  by  1,  we  have, 

S:  1  ::100  :  as,        .-.   S  =  —  • 

x 

A  table  may  be  computed  to  save  the  necessity  of  per 
forming  the  division. 

Densimeter. 

170.     The   densimeter  is  a  modification  of  the  volum- 
eter, and  admits  of  use  when  only  a  small  portion  of  the 
liquid  can  be  had,  as  is  often  the  case  in  examining 
animal  secretions,   such  as  bile,  chyle,  &c.     The  S 

construction  of  the  densimeter  differs  from  that  of 
the  volumeter,  last  described,  in  having  a  small 
cup  at  the  upper  extremity  of  the  stem,  destined 
to  receive  the  fluid  whose  specific  gravity  is  to  be 
determined. 

The  instrument  is  ballasted  so  that  when  the  cup 
is  empty,  the  densimeter  will  sink  in  distilled  water 
to  a  point  J5,  near  the  bottom  of  the  stem.  This 
point  is  the  0  of  the  instrument.  The  cup  is  then 
filled  with  distilled  water,  and  the  point  C,  to  Fig.  150 
which  it  sinks,  is  marked;  the  space  BC,  is  divi- 
ded into  any  number  of  equal  parts,  say  10,  and  the  grad- 
uation is  continued  to  the  top  of  the  tube 

To  use  the  instrument,  place  if  in  distilled  water,  and  fill 
the  cup  with  the  liquid  in  question,  and  note  the  division  to 
which  it  sinks.  Divide  10  by  the  number  of  this  division, 
and  the  quotient  will  be  the  specific  gravity  requ'red.  The 
principle  of  the  densimeter  is  the  same  as  that  of  the  volu- 
meter. 


MECHANICS    OF    LIQUIDS.  261 

Centesimal  Alcoholometer  of  Gay  Lussac. 

171.  This  instrument  is  the  same  in  construction  as  the 
scale  areometer ;  the  graduation  is,  however,  made  on  a  diff- 
erent principle.  Its  object  is,  to  determine  the  percentage  <>t 
alcohol  in  a  mixture  of  alcohol  and  water.  The  graduation  is 
made  as  follows :  the  instrument  is  first  placed  in  absolute 
alcohol,  and  ballasted  so  that  it  will  sink  nearly  to  the  top 
of  the  stem.  This  point  is  marked  1 00.  Next,  a  mixture 
of  95  parts  of  alcohol  and  5  of  water,  is  made,  and  the  point 
to  which  the  instrument  sinks,  is  marked  95.  The  inter- 
mediate space  is  divided  into  5  equal  parts.  Next,  a  mix- 
ture of  90  parts  of  alcohol  and  10  of  water  is  made;  the 
point  to  which  the  instrument  sinks,  is  marked  90,  and  the 
space  between  this  and  95,  is  divided  into  5  equal  parts.  In 
this  manner,  the  entire  stem  is  graduated  by  successive 
operations.  The  spaces  on  the  scale  are  not  equal  at  differ- 
ent points,  but,  for  a  space  of  five  parts,  they  may  be  re- 
garded as  equal,  without  sensible  error. 

To  use  the  instrument,  place  it  in  the  mixture  of  alcohol 
and  water,  and  read  the  division  to  which  it  sinks  ;  this  will 
indicate  the  percentage  of  alcohol  in  the  mixture. 

In  all  of  the  instruments,  the  temperature  has  to  be  taken 
into  account ;  this  is  usually  effected  by  means  of  correc- 
tions, which  are  tabulated  to  accompany  the  different 
instruments. 

On  the  principle  of  the  alcoholometer,  are  constructed  a 
great  variety  of  areometers,  for  the  purpose  of  determining 
the  degrees  of  saturation  of  wines,  syrups,  and  other  liquids 
employed  in  the  arts. 

In  some  nicely  constructed  hydrometers,  the  mercury 
used  as  ballast  serves  also  to  fill  the  bulb  of  a  delicate  ther- 
mometer, whose  stem  rises  into  the  cylinder  of  the  instru- 
ment, and  thus  enables  us  to  note  the  temperature  of  the 
fluid  in  which  it  is  immersed. 

EXAMPLES. 

1.  A  cubic  foot  of  water  weighs  1000  ounces.     Required 


262  MECHANICS. 

the  weight  of  a  cubical  block  of  stone,  one  of  whose  edges 
is  4  feet,  its  specific  gravity  being  2.5.  Ans.   10000  lbs. 

2.  Required  the  number  of  cubic  feet  in  a  body  whose 
weight  is  1000  lbs.,  its  specific  gravity  being  1.25. 

Ans.  12.8. 

3.  Two  lumps  of  metal  weigh  respectively  3  lbs.,  and  1  lb., 
and  their  specific  gravities  are  5  and  9.  What  will  be  the 
specific  gravity  of  an  alloy  formed  by  melting  them  together, 
supposing  no  contraction  of  volume  to  take  place. 

Ans.    5.625. 

4.  A  body  weighing  20  grains  has  a  specific  gravity  of  2.5. 
Required  its  loss  of  weight  in  water.  Ans.  8  grains. 

5.  A  body  weighs  25  grains  in  water,  and  40  grains  in  a 
liquid  whose  specific  gravity  is  .7.  What  is  the  weight  of 
the  body  in  vacuum  ?  Ans.  75  grains. 

6.  A  Nicholson's  hydrometer  weighs  250  grains,  and  it 
requires  an  additional  weight  of  336  grains  to  sink  it  to  the 
notch  in  the  stem,  in  a  mixture  of  alcohol  and  water.  What 
is  the  specific  gravity  of  the  mixture?  Ans.  .781. 

7.  A  block  of  wood  is  found  to  sink  in  distilled  water  till 
•£  of  its  volume  is  submerged.     What  is  its  specific  gravity  ? 

Ans.     .875. 

8.  The  weight  of  a  piece  of  cork  in  air,  is  f  oz. ;  the 
weight  of  a  piece  of  lead  in  water,  is  6|  oz. ;  the  weight  of 
the  cork  and  lead  together  in  Mater,  is  4tJ-q  oz.  What  is 
the  specific  gravity  of  the  cork  ?  Ans,  0.24. 

9.  A  solid,  whose  weight  is  250  grains,  weighs  in  water, 
147  grains,  and,  in  another  fluid,  120  grains.  What  is  the 
specific  gravity  of  the  latter  fluid  ?  Ans.   1.26°. 

10.  A  solid  weighs  60  grains  in  air,  40  in  water,  and  30  in 
an  acid.     What  is  the  specific  gravity  of  the  acid  ? 

Ans.  1.5. 


MECHANICS    OF    LIQUIDS. 


:63 


The  following  table  of  the  specific  gravity  of  some  of  the 
most  important  solid  and  fluid  bodies,  is  compiled  from  a 
table  given  in  the  Ordnance  Manual. 

TABLE    OF    SPECIFIC    GRAVITIES    OF    SOLIDS    AND   LIQUIDS. 


SOLIDS. 

SPEC.  GKAV. 

80LIDS. 

SPEC.  6EAV. 

Antimony,  cast 

6.712 

8.396 

8  788 

19.361 

7.788 

7.207 

11.352 

13.598 

13.580 

22.069 

20.337 

10.511 

7.291 

6.861 

1.900 

2.784 

1.270 

3.521 

1  500 

2.168 

1.822 

Limestone 

Marble,  common .... 

Salt,  common 

Sand 

Slate 

Stone,  common 

Tallow 

Boxwood 

3.180 
2.686 

Copper,  cast 

Gold,  hammered 

Iron,  bar 

Iron,  cast 

2.130 
1.800 

2.672 
2.520 

0 .  945 

Mercury  at  32°  F 

"        at  60° 

Platina,  rolled 

"       hammered. . . 
Silver,  hammered. . . . 

0.912 

Cedar 

Cherry 

Lignum  vitae 

Mahoganv 

0 .  596 
0.715 
1.333 

<>.8o4 

Tin,  cast 

1.170 

Zinc,  cast 

Bricks 

Chalk 

Coal,  bitumiuous 

Diamond 

Earth,  common 

Gvpsum 

Pine,  yellow 

Nitric  acid 

Sulphuric  acid 

Alcohol,  absolute...  . 
Ether,  sulphuric  .... 

Sea  water 

Olive  oil 

0.660 
1.217 
1.841 
0.792 
0.715 
1.026 
0.915 

Ivory  

Oil  of  Turpentine  . .  . 

0.870 

Thermometer. 

172.  A  thermometer  is  an  instrument  used  for  measur- 
ing the  temperatures  of  bodies.  It  is  found,  by  observation, 
that  almost  all  bodies  expand  when  heated,  and  contract 
when  cooled,  so  that,  other  things  being  equal,  they  always 
occupy  the  same  volumes  at  the  same  temperatures.  It  is 
also  found  that  different  bodies  expand  and  contract  in  a 
different  ratio  for  the  same  increments  of  temperature.  As 
a  general  rule,  liquids  expand  much  more  rapidly  than  solids, 
and  gases  much  more  rapidly  than  liquids.  The  construc- 
tion of  the  thermometer  depends  upon  this  principle  of 
unequal  expansibility  of  different  bodies.  A  great  variety 
of  combinations  have  been  used  in  the  construction  of  ther- 


:    as 


J 


Fig.  151. 


264:  MECHANICS. 

mometers,  only  one  of  which,  the  common  mercurial  ther 
mometer,  will  be  described. 

The  mercurial  thermometer  consists  of  a  cylindrical  or 
spherical  bulb  A,  at  the  upper  extremity  of  which, 
is  a  narrow  tube  of  uniform  bore,  hermetically 
sealed  at  its  upper  end.  The  bulb  and  tube  are 
nearly  filled  with  mercury,  and  the  whole  is 
attached  to  a  frame,  on  which  is  a  scale  for  deter- 
mining the  temperature,  which  is  indicated  by  the 
rise  and  fall  of  the  mercury  in  the  tube. 

The  tube  should  be  of  uniform  bore  through- 
out, and,  when  this  is  the  case,  it  is  found  that 
the  relative  expansion  of  the  mercury  and  glass 
is  very  nearly  uniform  for  constant  increments  of 
temperature.  A  thermometer  maybe  constructed 
and  graduated  as  follows :  A  tube  of  uniform 
Lore  is  selected,  and  upon  one  extremity  a  bulb  is 
blown,  which  may  be  cylindrical  or  spherical ;  the  former 
shape  is,  on  many  accounts,  the  preferable  one.  At  the 
other  extremity,  a  conical-shaped  funnel  is  blown  open  at 
the  top.  The  funnel  is  filled  with  mercury,  which  should  be 
of  the  purest  quality,  and  the  whole  being  held  vertical,  the 
heat  of  a  spirit-lamp  is  applied  to  the  bulb,  which  expand- 
ing the  air  contained  in  it,  forces  a  portion  in  bubbles  up 
through  the  mercury  in  the  funnel.  The  instrument  is  next 
allowed  to  cool,  when  a  portion  of  mercury  is  forced  down 
the  capillary  tube  into  the  bulb.  By  a  repetition  of  this 
process,  the  entire  bulb  may  be  filled  with  mercury,  as  well 
as  the  tube  itself.  Heat  is  then  applied  to  the  bulb,  until 
the  mercury  is  made  to  boil ;  and,  on  being  cooled  down  to 
a  little  above  the  highest  temperature  which  it  is  desired  to 
measure,  the  top  of  the  tube  is  melted  off  by  means  of  a 
jet  of  flame,  urged  by  a  blow-pipe,  and  the  whole  is  her- 
metically sealed.  The  instrument,  thus  prepared,  is  attached 
to  a  frame,  and  graduated  as  follows: 

The  instrument  is  plunged  into   a  bath  of  melting  ice, 
and,  after  being  allowed  to  remain  a  sufficient  time  for  the 


MECHANICS    OF   LIQUIDS.  2G5 

parts  of  the  instrument  to  take  the  uniform  temperature  of 
the  melting  ice,  the  height  of  the  mercury  in  the  tube  is 
marked  on  the  scale.  This  gives  the  freezing  point  of  the 
scale.  The  instrument  is  next  plunged  into  a  bath  of  boiling 
water,  and  allowed  to  remain  long  enough  for  all  of  the  parts 
to  acquire  the  temperature  of  the  water  and  steam.  The 
height  of  the  mercury  is  then  marked  on  the  scale.  This 
gives  the  boiling  point  of  the  scale.  The  freezing  and 
boiling  points  having  been  determined,  the  intermediate 
space  is  divided  into  a  certain  number  of  equal  parts, 
according  to  the  scale  adopted,  and  the  graduation  is  then 
continued,  both  upwards  and  downwards,  to  any  desired 
extent.  Three  principal  scales  are  used.  Fahrenheit's 
scale,  in  which  the  space  between  the  freezing  and  boiling 
point  is  divided  into  ISO  equal  parts,  called  degrees,  the 
freezing  point  being  marked  32°,  and  the  boiling  point  212°. 
In  this  scale,  the  0  point  is  32  degrees  below  the  freezing 
point.  The  Centigrade  scale,  in  which  the  space  between 
the  fixed  points  is  divided  into  100  equal  parts,  called 
degrees.  The  0  of  this  scale  is  at  the  freezing  point. 
Reaumur's  scale,  in  which  the  same  space  is  divided  into 
80  equal  parts,  called  degrees.  The  0  of  this  scale  also  is 
at  the  freezing  point. 

If  we  denote  the  number  of  degrees  on  the  Fahrenheit, 
Centigrade,  and  Reaumur  scales,  by  F,  C,  and  R  respec- 
tively, the  following  formula  will  enable  us  to  j:>ass  from 
any  one  of  these  scales  to  any  other  : 

±{F°  -32)  =  \C°  =  Ji2°. 

The  scale  most  in  use  in  this  country  is  Fahrenheit's 
The  other  two  are  much  used  in  Europe,  particularly  the 
Centigrade  scale. 

Velocity  of  a  liquid  flowing  through  a  small  orifice. 

173.     Let  ABD  represent  a  vessel,  having  a  very  small 
orifice  at  its  bottom,  and  filled  with  any  liquid. 
12 


266 


MECHANICS. 


Denote  the  area  of  the  orifice,  by  a,  and  its 
depth  below  the  upper  surface,  by  h.  Let  D 
represent  an  infinitely  small  layer  of  the  liquid 
situated  nt  the  orifice,  and  denote  its  height, 
by  h '.  This  layer  is  (Art.  155)  urged  down- 
wards by  a  force  equal  to  the  weight  of  a 
column  of  the  liquid  whose  base  is  equal  to  the  orifice,  and 
whose  height  is  h ;  denoting  this  pressure,  by  p,  and  the 
weight  of  a  unit  of  volume  of  the  liquid,  by  10,  we  shall 
have, 

p  =  wah. 


If  the  element  is  pressed  downwards  by  its   own  weight 
alone,  this  pressure  being  denoted  by^',  we  have, 

p   —  wah!. 


Dividing  the  former  equation   by  the  latter,  member   by 
member,  we  have, 

p       h 

p'  ~  A,; 

that  is,  the  pressures  are  to  each  other  as  the  heights  h 
and  h'. 

AVere  the  element  to  fall  through  the  small  height  h\ 
under  the  action  of  the  pressure^',  or  its  own  weight,  the 
velocity  generated  would  (Art.  115)  be  given  by  the 
equation, 


v'  -.=  y2gh'. 


Denoting  the  velocity  actually  generated  whilst  the  ele- 
ment is  falling  throught  the  height  h\  by  r,  and  recol- 
lecting that  the  velocities  generated  in   falling  through   a 


MECHANICS    OF    LIQUIDS.  267 

given  height,  are  to  each  other  as  the  square  roots  of  the 
pressures,  we  shall  have, 


v  :  v'  :  :   y ' p  :   V^/,         .'.     v  =  v'  \J ~,  • 
Substituting  for  v'  its  value,  just  deduced,   and  for  — ,  its 

h  p 

value,   — ,  we  have 

(150.) 


Hence,  we  conclude  that  a  liquid  loill  issue  from  a  very 
small  orifice  at  the  bottom  of  the  containing  vessel,  with  a 
velocity  equal  to  that  acquired  by  a  heavy  body  in  falling 
freely  through  a  height  equal  to  the  depth  of  the  orifice 
below  the  surface  of  the  fluid. 

We  have  seen  that  the  pressure  due  to  the  weight  of  a 
fluid  upon  any  point  of  the  surface  of  the  containing  vessel, 
is  normal  to  the  surface,  and  is  always  proportional  to  the 
depth  of  the  point  below  the  level  of  the  free  surface. 
Hence,  if  the  side  of  a  vessel  be  thin,  so  as  not  to  affect  the 
flow  of  the  liquid,  and  an  orifice  be  made  at  any  point,  the 
liquid  will  flow  out  in  a  jet,  normal  to  the  surface  at  the 
opening,  and  with  a  velocity  due  to  a  height  equal  to  that 
of  the  orifice  from  the  free  surface  of  the  fluid. 

If  the  orifice  is  on  the  vertical  side  of  a  vessel,  the  initial 
direction  of  the  jet  will  be  horizontal ;  if  it  be  made  at  a 
point  where  the  tangent  plane  is  oblique  to  the  horizon,  the 
initial  direction  of  the  jet  will  be  oblique ;  if  the  opening  is 
made  on  the  upper  side  of  a  por- 
tion of  a  vessel  where  the  tangent 
is  horizontal,  the  jet  will  be 
directed  upwards,  and  will  rise 
to  a  height  due  to  the  velocity ; 
that   is,   to    the   height    of   the  Fig.  158. 

upper  surface  of  the  fluid.     This 


T 

"""%B 

Df- 

<'  Y" 

-^0 

268  MECHANIC?. 

can  be  illustrated  experimentally,  by  introducing  a  tube  near 
the  bottom  of  a  vessel  of  water,  and  bending  its  outer 
extremity  upwards,  when  the  fluid  will  be  observed  to  rise 
to  the  level  of  the  upper  surface  of  the  water  in  the  vessel. 

Spouting  of  Liquids  on  a  Horizontal  Plane. 

174.  Let  KL  represent  a  vessel  lilled  with  water.  Let 
D  represent  an  orifice  in  its  ver- 
tical side,  and  BE  the  path 
described  by  the  spouting  fluid. 
We  may  regard  each  drop  of 
water  as  it  issues  from  the  orifice, 
as  a   projectile   shot  forth   hori-  fcSl-— —^'jl7 

zontally,  and  then  acted  upon  by  ^^   ■'''' ~^~iy 

the   force  of  gravity.     Its  path  «    ^ 

will,    therefore,   be   a   parabola, 

and  the  circumstances  of  its  motion  will  be  made  known  by 

a  discussion  of  Equations  (115)  and  (120). 

Denote  the  distance  BK,  by  A',  and  the  distance  BL,  by 
h.  TVe  have,  from  Equation  (120),  by  making  y  equal  to 
h\  and  x  =  KE, 


■*„ 


KE  = 


9 


But  we  have  found  that  v  =  V  2gh ;    hence,  by  substitu- 
tion, we  have, 

KE  =  2V  M7- 

If  we  describe  a  semicircle  on  EL,  as  a  diameter,  and 
through  D  draw  an  ordinate  BIT,  we  shall  have,  from  a 
well-known  property  of  the  circle, 


BII  =  y/BK.BL  = 
Hence  we  have,  by  substitution, 

KE  =  2BH. 


MECHANICS    OF    LIQUIDS.  269 

Since  there  are  two  points  on  KL  at  which  the  ordinates 
are  equal,  it  follows  that  there  are  two  oririces  through 
which  the  fluid  will  spout  to  the  same  distance  on  the 
horizontal  plane ;  one  of  these  will  be  as  far  above  the 
centre  0,  as  the  other  is  below  it. 

If  the  orifice  be  at  0,  midway  between  K  and  Z,  the 
ordinate  OS  will  be  the  greatest  possible,  and  the  range 
KE'  will  be  a  maximum.  The  range  in  this  case  will  be 
equal  to  the  diameter  of  the  circle  LHK,  or  to  the 
distance  from  the  level  of  the  water  in  the  vessel  to  the 
horizontal  plane. 

If  a  semi-parabola  EE'be  described,  having  its  axis  ver- 
tical, its  vertex  at  X,  and  focus  at  iT,  then  may  every  point 
P,  within  the  curve,  be  reached  by  two  separate  jets  issuing 
from  the  side  of  the  vessel ;  every  point  on  the  curve  can  be 
reached  by  one,  and  only  one ;  whilst  points  lying  without 
the  curve  cannot  be  reached  by  any  jet  whatever. 

If  the  jet  is  directed  obliquely  upwards  by  a  short  pipe 
A  (Fig.  153),  the  path  described  by  each  particle  will  still  be 
the  arc  of  a  parabola  ABC.  Since  each  particle  of  the 
liquid  may  be  regarded  as  a  body  projected  obliquely  up- 
ward, the  nature  of  the  path  and  the  circumstances  of  the 
motion  will  be  given  by  Equation  ( 115 ). 

In  like  manner,  a  discussion  of  the  same  equation  will 
make  known  the  nature  of  the  path  and  the  circumstances 
of  motion,  when  the  jet  is  directed  obliquely  downwards  by 
means  of  a  short  tube. 

Modifications  due  to  extraneous  pressure. 

175.  If  we  suppose  the  upper  surface  of  the  liquid,  in 
any  of  the  preceding  cases,  to  be  pressed  by  any  force,  as 
when  it  is  urged  downwards  by  a  piston,  we  may  denote  the 
height  of  a  column  of  fluid  whose  weight  is  equal  to  the  ex- 
traneous pressure,  by  h '.  The  velocity  of  efflux  will  then  be 
given  by  the  equation, 


v  =   V2ff(A  +  h')< 


270  MECHANICS. 

The  pressure  of  the  atmosphere  acts  equally  on  the  upper 
surface  and  the  surface  of  the  opening ;  hence,  in  ordinary 
cases,  it  may  be  neglected ;  but  were  the  water  to  flow  into 
a  vacuum,  or  into  rarefied  air,  the  pressure  must  be  taken 
into  account,  and  this  may  be  done  by  means  of  the  formula 
just  given. 

Should  the  flow  take  place  into  condensed  air,  or  into  any 
medium  which  opposes  a  greater  resistance  than  the  atmos- 
pheric pressure,  the  extraneous  pressure  would  act  upwards, 
ti  would  be  negative,  and  the  preceding  formula  would 
become, 

v  =   \^2g(h  —  A'), 

Coefficients  of  Efflux  and  Velocity. 

176.  When  a  vessel  empties  itself  through  a  small  orifice 
at  its  bottom,  it  is  observed  that  the  particles  of  fluid  near 
the  top  descend  in  vertical  lines;  when  they  approach  the 
bottom  they  incline  towards  the  orifice,  the  converging  lines 
of  fluid  particles  tending  to  cross  each  other  as  they  emerge 
from  the  vessel.  The  result  is,  that  the  stream  grows  nar- 
rower, after  leaving  the  vessel,  until  it  reaches  a  point  at  a 
distance  from  the  vessel  equal  to  about  the  radius  of  the 
orifice,  when  the  contraction  becomes  a  minimum,  and  below 
that  point  the  vein  again  spreads  out.  This  phenomenon  is 
called  the  contraction  of  the  vein.  The  cross  section  at  the 
most  contracted  part  of  the  vein,  is  not  far  from  T\\  of  the 
area  of  the  orifice,  when  the  vessel  is  very  thin.  If  we  de- 
note the  area  of  the  orifice,  by  a,  and  the  area  of  the  least 
cross  section  of  the  vein,  by  a',  we  shall  have, 

a'  =  ka, 

in  which  k  is  a  number  10  be  determined  by  experiment. 
This  number  is  called  the  coefficient  of  contraction. 

To  find  the  quantity  of  water  discharged  through  an  ori- 
fice at  the  bottom  of  the  containing  vessel,  in  a  second,  we 
have  only  to  multiply  the  area  of  the  smallest  cross  section 


MECHANICS    OF    LIQUIDS.  271 

of  the  vein,  by  the  velocity.     Denoting  the  quantity  dis- 
charged in  one  second,  by  Q\  we  shall  have, 

Q'  =  ha  \/2(/h. 

This  formula  is  only  true  on  the  supposition  that  the 
actual  velocity  is  equal  to  the  theoretical  velocity,  which  is 
not  the  case,  as  has  been  shown  by  experiment.  The  theo- 
retical velocity  has  been  shown  to  be  equal  to  y^A,  and 
if  we  denote  the  actual  velocity,  by  v\  we  shall  have, 


in  which  I  is  to  be  determined  by  experiment ;  this  value  of 
I  is  slightly  less  than  1,  and  is  called  the  coefficient  of  veloc- 
ity. In  order  to  get  the  actual  discharge,  we  must  replace 
■\/2gh  by  l\^2(/h,  in  the  preceding  equation.  Doing  so, 
and  denoting  the  actual  discharge  per  second,  by  Q,  Ave  have. 


The  product  hi,  is  called  the  coefficient  of  efflux.  It  has 
been  shown  by  experiment,  that  this  coefficient  for  orifices 
in  thin  plates,  is  not  quite  constant.  It  decreases  slightly,  as 
the  area  of  the  orifice  and  the  velocity  are  increased  ;  and 
it  is  further  found  to  be  greater  for  circular  orifices  than  for 
those  of  any  other  shape. 

If  we  denote  the  coefficient  of  efflux,  by  ?n,  we  have, 


In  this  equation,  h  is  called  the  head  of  water.  Hence, 
we  may  define  the  head  of  water  to  be  the  distance  from 
the  orifice  to  the  piano,  of  the  upper  surface  of  the  fluid. 

The  mean  value  of  m  corresponding  to  orifices  of  from 
|  to  6  inches  in  diameter,  with  from  4  to  20  feet  head  of 


272 


MKCIIANICS. 


water,  has  been  found  to  be  about   .615.     If  we  take  the 
value  of  k  =  .64,  we  shall  have, 

m        .615 
1  =  k    =  MO  =  ^ 

That  is,  the  actual  velocity  is  only  T9y6y  of  the  theoretical 
velocity.     This  diminution  is  due  to  friction,  viscosity,  <fce. 

Efflux  through  Short  Tubes. 

177.  It  is  found  that  the  discharge  from  a  given  orifice 
is  increased,  when  the  thickness  of  the  plate  through  which 
the  flow  takes  place,  is  increased  ;  also,  when  a  short  tube  is 
introduced. 

When  a  tube  AB,  is  employed  which  is  not  more  than 
four  times  as  long  as  the  diameter  of  the 
orifice,  the  value  of  m  becomes,  on  an  aver- 
age, equal  to  .813;  that  is,  the  discharge 
per  second  is  1.325  times  greater  when  the 
tube  is  used,  than  without  it.  In  using  the 
cylindrical  tube,  the  contraction  takes  place 
at  the  outlet  of  the  vessel,  and  not  at  the  outlet  of  the  tube. 

Compound  mouth-pieces  are  sometimes  used  formed  of 
two  conic  frustrums,  as  shown  in  the  figure, 
having  the  form  of  the  vein.     It  has  been 
shown   by  Etelwein,  that  the  most  effec-  J^.jj/ST" 

tive    tubes  of  this   form  should   have  the 
diameter    of  the   cross   section   CD,  equal  ^^fvE 

to  .8:13   of  the  diameter  AB.     The  angle  Pig.156. 

made  by  the  sides  CE  and  DF,  should  be 
ah   ut  5°  9',  and  the  length  of  this  portion  should  be  throe 
times  that  of  the  other. 

EXAMPLES. 

1.  With  what  theoretical  velocity  will  water  issue  from  a 
small  orifice  16^  feet  below  the  surface  of  the  fluid  ? 

Ans.  32£  ft, 


MECHANICS    OF    LIQUIDS.  273 

2.  If  the  area  of  the  orifice,  in  the  last  example,  is  ^  of  a 
square  foot,  and  the  coefficient  of  efflux  .615,  how  many- 
cubic  feet  of  water  will  be  discharged  per  minute  ? 

Ans.    118.695  ft. 

3.  A  vessel,  constantly  filled  with  water,  is  4  feet  high, 
with  a  cross-seclion  of  one  square  foot ;  an  orifice  in  the 
bottom  has  an  area  of  one  square  inch.  In  what  time  will 
three-fourths  of  the  water  be  drawn  off,  the  coefficient  of 
efflux  being  .6  ?  Ans,  \  minute,  nearly. 

4.  A  vessel  is  kept  constantly  full  of  water.  How  many 
cubic  feet  of  water  will  be  discharged  per  minute  from  an 
orifice  9  feet  below  the  upper  surface,  having  an  area  of  1 
square  inch,  the  coefficient  of  efflux  being  .6  ? 

Ans.  6  cubic  feet,  about. 

5.  In  the  last  example,  what  will  be  the  discharge  per 
minute,  if  we  suppose  each  square  foot  of  the  upper  surface 
to  be  pressed  by  a  force  of  645  lbs.  ? 

Ans.   8-J  cubic  feet,  about. 

6.  The  head  of  water  is  16  feet,  and  the  orifice  is  T|T  of 
a  square  foot.  What  quantity  of  water  will  be  discharged 
per  second,  when  the  orifice  is  through  a  thin  plate  ? 

solution. 
In  this  case,  we  have, 


Q  =  .615  x  .01^/2  X  321  x  16  =  .197  cubic  feet. 

When  a  short  cylindrical  tube  is  used,  we  have, 

Q  =  .197  x  1.325  =  .261  cubic  feet. 

In  Etelweix's  compound  mouthpiece,  if  we  take  the 
smallest  cross-section  as  the  orifice,  and  denote  it  by  a,  it  is 
found  that  the  discharge  is  2£  times  that  through  an  orifice 
of  the  same  size  in  a  thin  plate.  In  this  case,  we  have,  sup- 
posing a  =  ji-Q  of  a  square  foot, 

Q  =  .197  y  g£  =?,,49  cubic  feet. 
12* 


274  MECHANICS. 

Motion  of  water  in  open  channels. 

17§.  When  water  flows  through  an  open  channel,  as  in 
A  river,  canal,  or  open  aqueduct,  the  form  of  the  channel 
being  always  the  same,  and  the  supply  of  water  being  con- 
stant, it  is  a  matter  of  observation  that  the  flow  becomes 
uniform ;  that  is,  the  quantity  of  water  that  flows  through 
any  cross-section,  in  a  given  time,  is  constant.  On  account 
of  adhesion,  friction,  <fcc.,  the  particles  of  water  next  the 
sides  and  bottom  of  the  channel  have  their  motion  retarded. 
This  retardation  is  imparted  to  the  next  layer  of  particles, 
but  in  a  less  degree,  and  so  on,  till  a  line  of  particles  is 
reached  whose  velocity  is  greater  than  that  of  any  other 
filament.  This  line,  or  filament  of  particles,  is  called  the 
axis  of  the  stream.  In  the  case  of  cylindrical  pipes,  the 
axis  coincides  sensibly  with  the  axis  of  the  pipe ;  in  straight, 
open  channels,  it  coincides  with  that  line  of  the  upper  sur- 
face which  is  midway  between  the  sides. 

A  section  at  right-angles  to  the  axis  is  called  a  cross-sec- 
Hon,  and,  from  what  has  been  shown,  the  velocities  of  the 
fluid  particles  will  be  different  at  different  points  of  the 
same  cross-section.  The  mean  velocity  corresponding  to 
any  cross-section,  is  the  average  velocity  of  the  particles  at 
every  point  of  that  section.  The  mean  velocity  may  be 
found  by  dividing  the  volume  which  flows  through  the  sec- 
tion in  one  second,  by  the  area  of  the  cross-section.  Since 
the  same  volume  flows  through  each  cross-section  per 
second,  after  the  flow  has  become  uniform,  it  follows  that, 
in  channels  of  varying  width,  the  mean  velocity,  at  any 
section,  will  be  inversely  as  the  area  of  the  section. 

The  intersection  of  the  plane  of  cross-section  with  the 
sides  and  bottom  of  the  channel,  is  called  the  perimeter  of 
the  section.  In  the  case  of  a  pipe  which  is 'constantly  tilled, 
the  perimeter  is  the  entire  line  of  intersection  of  the  plane 
of  cross-section,  with  the  interior  surface  of  the  pipe. 

The  mean  velocity  of  water  in  an  open  channel  depends, 
in  the  first  place,  upon  its  inclination  to  the  horizon.  As  the 
inclination  becomes  greater,  the  component  of  gravity  in  the 


MECHANICS    OF    LIQUIDS.  275 

direction  of  the  channel  increases,  and,  consequently,  the 
velocity  becomes  greater.  Denoting  the  inclination  by  7",  and 
resolving  the  force  of  gravity  into  two  components,  one  at 
right  angles  to  the  upper  surface,  and  the  other  parallel  to 
it,  we  shall  have  for  the  latter  component, 

gsinl. 

This  is  the  only  force  that  acts  to  increase  the  velocity. 
The  velocity  will  be  diminished  by  friction,  adhesion,  &c. 
The  total  effect  of  these  resistances  will  depend  upon  the 
ratio  of  the  perimeter  to  the  area  of  the  cross  section,  and 
also  upon  the  velocity.  The  cross-section  being  the  same, 
the  resistances  will  increase  as  the  perimeter  increases ;  con- 
sequently, for  the  same  cross-section,  the  resistance  of  fric- 
tion will  be  the  least  possible  when  the  perimeter  is  least 
possible.  The  retardation  of  the  flow  will  also  diminish  as 
the  area  of  the  cross-section  is  increased,  other  things  re- 
maining unchanged. 

If  we  denote  the  area  of  the  cross-section  by  a,  the 
perimeter,  by  J?,  and  the  velocity,  by  v,  we  shall  have, 

—p-  =  /Mi 

in  which  f  denotes  some  function  of  v. 

Since  the  inclination  is  very  small  in  all  practical  cases, 
we  may  place  the  inclination  itself  for  the  sine  of  the  inclin- 
ation, and  doing  so,  it  has  been  shown  by  Peony,  that  the 
function  of  v  may  be  expressed  by  two  terms,  one  of  which 
is  of  the  first,  and  the  other  of  the  second  degree,  with  re- 
spect to  v ;  or, 

gal  ,      o 


Denoting  —  by  JR,         by  ft,   and   -  by  I,  we  have,  finally, 
kv  +  lv2  =  HI. 


276  MECHANICS. 

in  which  k  and  I  are  constants,  to  be  determined  by  experi- 
ment.    According  to  Etelweix,  we  have, 

h   -   .0000242651,     and    I  =   .0001114155. 

Substituting  these  values,  and  solving  with  respect  to  y, 
we  have. 


V   =    —  0.10S8941G04  +   ^.0118580490  +  8975.41 4285^1, 

from  which  the  velocity  can  be  found  when  H  and  I  are 
known.  The  values  of  k  and  /,  and  consequently  that  of  v, 
were  found  by  Peony  to  be  somewhat  different  from  those 
given  above.  Those  of  Etelweix  are  selected  for  the  reason 
that  they  were  based  upon  a  much  larger  number  of  exper- 
iments than  those  of  Phony. 

Having  the  mean  velocity  and  the  area  of  the  cross- sec- 
tion, the  quantity  of  water  delivered  in  any  time  can  be 
computed.  Denoting  the  quantity  delivered  in  n  seconds, 
by  Q,  and  retaining  the  preceding  notation,  we  have, 

Q  =  nor. 

The  quantity  of  water  to  be  delivered  is  generally  one  of 
the  data  in  all  practical  problems  involving  the  distribution 
of  water.  The  difference  of  level  of  the  point  of  supply 
and  delivery  is  also  known.  The  preceding  principles  ena- 
ble us  to  give  such  a  form  to  the  cross-section  of  the  canal, 
or  aqueduct,  as  will  ensure  the  requisite  supply. 

Were  it  required  to  apply  the  results  just  deduced,  to  the 
rase  of  irregular  channels,  or  to  those  in  which  there  were 
man)  curves,  a  considerable  modification  would  be  required. 
The  theory  of  these  modifications  does  not  come  within  the 
limits  assigned  to  this  treatise.  For  a  complete  discussion 
of  the  whole  subject  of  hydraulics  in  a  popular  form,  the 
reader  is  referred  to  the  Traiti  </'  Hydra ulique  D'Auhebsok 


MECHANICS    OF    LIQUIDS.  277 

Motion  of  water  in  pipes. 

179.     The  circumstances  of  the  motion  of  water  in  pipes, 
are  closely  analagous  to  those  of  its 
motion    in     open     channels.      The       , — i 

forces  which  tend  to  impart  motion       jjg|  

are  dependent  upon  the  weight  of  ^"^^5"""^^^ 
the  water  in  the.  pipe,  and  upon  the  ^N^Jgjj] 

height  of  the  water  in  the  upper  Fig.  15T. 

reservoir.     Those    which    tend    to 

prevent  motion  depend  upon  the  depth  of  water  in  the 
lower  reservoir,  friction  in  the  pipe,  adhesion,  and  shocks 
arising  from  irregularities  in  the  bore  of  the  pipe.  The  re- 
tardation due  to  shocks  will,  for  the  present,  be  neglected. 

Let  AB  represent  a  straight  cylindrical  pipe,  connecting 
two  reservoirs  Jl  and  H'.  Suppose  th#  water  to  maintain 
its  level  at  E,  in  the  upper,  and  at  (7,  in  the  lower  reservoir 
Denote  'AJB,  by  A,  and  BC,  by  h'.  Denote  the  length  of 
the  pipe,  by  I,  its  circumference,  by  c,  its  cross-section,  by 
a,  its  inclination,  by  9,  and  the  weight  of  a  unit  of  volume 
of  water,  by  10. 

Experience  shows  that,  under  the  circumstances  above 
indicated,  the  flow  soon  becomes  uniform.  We  may  then 
regard  the  entire  mass  of  fluid  in  the  pipe  as  a  coherent 
solid,  moving  with  a  mean  uniform  velocity  down  the 
inclined  plane  AB. 

The  weight  of  the  water  in  the  pipe  will  be  equal  to  waL 
If  we  resolve  this  weight  into  two  components,  one  perpen- 
dicular to,  and  the  other  coinciding  with  the  axis  of  the 
tube,  we  shall  have  for  the  latter  component,  loal sm$.  But 
/sins  is  equal  to  DB.  Denoting  this  distance  by  A",  wei 
shall  have  for  the  pressure  in  the  direction  of  the  axis,  duel 
to  the  weight  of  the  water  in  the  pipe,  the  expression  wah". 
This  pressure  acts  from  A  towards  B.  The  pressure  due  to 
the  weight  of  the  water  in  7i,  and  acting  in  the  same 
direction,  is  wah. 

The  forces  acting  from  B  towards  A,  are,  first,  that  d  ae 


278  1CE0HANIOB. 

to  the  weight  of  the  'water  in  It',  which  is  equal  to  wall'; 
and,  secondly,  the  resistance  due  to  friction  and  adhesion. 
This  resistance  depends  upon  the  length  of  the  pipe,  its 
circumference  and  the  velocity.  It  has  been  shown,  by 
experiment,  that  this  force  may  be  expressed  by  the  term, 

cl(kv  -(■  fcV). 

Since  the  velocity  has  been  supposed  uniform,  the  forces 
acting  in  the  direction  of  the  axis,  must  be  in  equilibrium. 
Hence, 

wall  4-  wah"  =  ioah'  4-  cl(kv  4-  k'v*) ; 

whence,  by  reduction, 

k  k'    _       a/h  4-  h"  —  h'\ 

—  v  +  —  V  =  -  ( • 

ic  w  c\  I  J 


The  factor  -  is  equal  to  one-fourth  of  the  diameter  of  the 

pipe.    Denoting  this  by  c7,  we  shall  have,   —  =  \d  ;   denot- 

•        *    i.  k'  i     '  a  h  +  h"-h'  \ 

ing  —    by  m,   —    by  w,    and  =■ by  s,   we   have, 

w  w  t 

mv  +  nv*  =  Jcfe. 


The  values  of  m  and  «,  as  determined  experimentally  by 
Pkont,  are, 

m  =  0.00017,      and     n  =  0.000106. 

Hence,  by  substitution, 

.0001 7v  4  .000106^'  =  J<fo. 

If  v  is  not  very  small,  the  first  term  may  be  neglected, 
which  will  give, 

V  —  48.56  y/ds. 


MECHANICS    OF    LIQUIDS. 


$79 


If  we  denote  the  quantity  of  water  delivered  in  n  sec- 
onds, by  Q,  we  shall  have, 

Q  —  nav  =  48.56?ia\/ds. 

The  velocity  will  be  greatly  diminished,  if  the  tube  is 
curved  to  any  considerable  extent,  or  if  its  diameter  is  not 
uniform  throughout.  It  is  not  intended  to  enter  into  a 
discussion  of  these  cases ;  their  complete  development 
would  require  more  space  than  has  been  allotted  to  this 
branch  of  Mechanics. 

General  Remarks  on  the  distribution  and  flow  of  water  in  pipes. 

180.  Whenever  an  obstacle  occurs  in  the  course  of  an 
open  channel  or  pipe,  a  change  of  velocity  must  take  place. 
In  passing  the  obstacle,  the  velocity  of  the  water  will  increase, 
and  then,  impinging  upon  that  which  has  already  passed,  a 
shock  will  take  place.  This  shock  consumes  a  certain 
amount  of  living  force,  and  thus  diminishes  the  velocity  of 
the  stream.  All  obstacles  should  be  avoided  ;  or,  if  any  are 
unavoidable,  the  stream  should  be  diminished,  and  again 
enlarged  gradually,  so  as  to  avoid,  as  much  as  possible,  the 
necessary  shock  incident  to  sudden  changes  of  velocity.    ' 

For  a  like  reason,  when  a  branch  enters  the  main  channel, 
it  should  be  made  to  enter  as  nearly  in  the  direction  of  the 
current  as  possible. 

All  changes  of  direction  give  rise  to  mutual  impacts 
amongst  the  particles,  and  the  more,  as  the  change  is  more 
abrupt.  Hence,  when  a  change  of  direction  is  necessary, 
the  straight  branches  should  be  made  tangential  to  the 
curved  portion. 

The  entrance  to,  and  outlet  from  a  pipe  or  channel,  should 
be  enlarged,  in  order  to  diminish,  as  much  as  possible,  the 
coefficients  of  ingress  and  egress. 

When  a  pipe  passes  over  uneven  ground,  sometimes  as- 
cending, and  sometimes  descending,  there  is  a  tendency  to 
a  collection  of  bubbles  of  air,  at  the  highest  points,  which 


280  MECHANICS. 

may  finally  come  to  act  as  an  impeding  cause  to  the  flow. 
There  should,  therefore,  be  suitable  pipes  inserted  at  the 
highest  points,  to  permit  the  confined  air  to  escape. 

Finally,  attention  should  be  given  to  the  form  of  the  cross- 
section  of  the  channel.  If  the  channel  is  a  pipe,  it  should  be 
made  cylindrical.  If  it  is  a  canal  or  open  aqueduct,  that 
form  should  be  given  to  the  perimeter  which  would  give 
the  greatest  cross-section,  and,  at  the  same  time,  conform  to 
the  necessary  conditions  of  the  structure.  The  perimeter  in 
open  channels  is  generally  trapezoidal,  from  the  necessity 
of  the  case  ;  and  it  should  be  remembered,  that  the  nearer 
the  form  approaches  a  semi-circle,  the  greater  will  be  the 
flow. 

Capillary  Phenomena. 

1§1.  When  a  liquid  is  in  equilibrium,  under  the  action 
of  its  own  weight,  it  has  been  shown  that  its  upper  surface 
is  level.  It  is  observed,  however,  in  the  neighborhood  of 
solid  bodies,  such  as  the  walls  of  a  containing  vessel,  that 
the  surface  is  sometimes  elevated,  and  sometimes  depressed, 
according  to  the  nature  of  the  liquid  and  solid  in  contact. 
These  elevations  and  depressions  result  from  the  action  of 
molecular  forces,  exerted  between  the  particles  of  the  liquid 
and  solid  which  are  in  contact ;  from  the  fact  that  they  are 
more  apparent  in  the  case  of  small  tubes,  of  the  diameter  of 
a  hair,  these  phenomena  have  been  called  capillary  phenom- 
ena, and  the  forces  giving  rise  to  them,  capillary  forces. 

These  forces  only  produce  sensible  efFects  at  extremely 
small  distances.  Clahlattt  has  shown,  that  when  the  inten- 
sity of  the  force  of  attraction  of  t  lie  particles  of  the  solid  for 
those  of  the  liquid,  exceeds  one-half  that  of  the  particles  of 
the  liquid  for  each  other,  the  liquid  will  be  elevated  about 
tic-  solid;  when  less,  it  will  be  depressed;  when  equal,  it 
will  neither  be  elevated  nor  depressed.  In  the  first  case,  the 
resultant  of  the  capillary  forces  is  a  force  of  capillary  attrac- 
tion ;  in  thes  cml  i  ase,  it  i<  a  force  of  capillary  repulsion  ; 
and  in  the  third  case,  the  capillar)'  forces  are  in  equilibrium. 

The  following  are  some  of  the  observed  effects  of  capillary 


MECHANICS    OF    LIQUIDS.  281 

action  :  When  a  solid  is  plunged  into  a  liquid  which  is 
capable  of  moistening  it,  as  when  wood  or  glass  is  plunged 
into  water,  the  surface  of  the  liquid  is  heaped  up  about  the 
solid,  taking  a  concave  form,  as  shown  in  Fig.  158. 

When  a  solid  is  plunged  into  a  liquid  which 
is  not  capable  of  moistening  it,  as  when 
glass  is  plunged  into  mercury,  the  surface 
of  the  liquid  is  depressed  about  the  solid, 
taking  a  convex  form,  as  shown  in  Fig.  159.  rig.  158. 

The  surface  of  the  liquid  in  the  neighbor- 
hoed  of  the  bounding  surfaces  of  the  con- 
taining vessel  takes  the  form  of  concavity         ^_ 
or  convexity,  according  as  the  material  of 
the  vessel  is  capable  of  being  moistened,  ,  Ficr  159 

or  not,  by  the  liquid. 

These  phenomena  become  more  apparent  when,  instead  of 
a  solid  body,  we  plunge  a  tube  into  a  liquid,  according  as  the 
material  of  the  tube  is,  or  is  not,  capable  of  being  moistened  by 
the  liquid,  the  liquid  will  rise  in  the  tube  or  be  depressed  in 
it.  When  the  liquid  rises  in  the  tube,  its  upper  surface 
takes  a  concave  shape ;  when  it  is  depressed,  it  takes  a  con- 
vex form.  The  elevations  or  depressions  increase  as  the  dia- 
meter of  the  tube  diminishes. 


Elevation  and  Depression  between  plates. 

182.  If  two  plates  of  any  substance  are  placed  parallel 
to  each  other,  it  is  found  that  the  laws  of  ascent  and  descent 
of  the  liquid  into  which  they  are  plunged,  are  essentially  the 
same  as  for  tubes.  For  example :  if  two  plates  of  glass 
parallel  to  each  other,  and  pretty  close  together,  arc  plunged 
into  water,  it  is  found  that  the  water  will  rise  between  then) 
to  a  height  which  is  inversely  proportional  to  their  dist- 
ance apart;  and  further,  that  this  height  is  equal  to  half  tlu1 
height  to  which  water  would  rise  in  a  glass  tube  whose 
internal  diameter  is  equal  to  the  distance  between  the 
plates. 


2S2  MECHANICS. 

If  the  same  plates  are  plunged  into  mercury,  there  v,  ill  be 
a  depression  according  to  an  analagous  law. 

If  two  plates  of  glass,  AB  and  A  C,  inclined  to  each  other, 
as  shown  in  Fig.  160,  their  line  of 
junction  being  vertical,  be  plunged  a  C 

into  any  liquid  which  will  moisten  B 

them,  the    liquid  will  rise  between  JS?*.. 

them.     It  will  rise  higher  near  the 
junction,  the  surface  taking  a  curved 

form,  such  that  any  section  made  by  a  plane  through  A, 
will  be  an  equilateral  hyperbola.  This  form  of  the  elevated 
fluid  conforms  to  the  laws  above  explained. 

If  the  line  of  junction  of  the  two  plates  is 
horizontal,  a  small  quantity  of  a  liquid  between 
them,  which  will  moisten  them,  will  assume 
the  shape  shown  at  A.      If  the   liquid  does  Fig.  i6i. 

not  moisten  the  plates,  it  will  take  the  form 
shown  at  B. 

Attraction  and  Repulsion  of  Floating  Bodies. 

183.  If  two  small  balls  of  wood,  both  of  which  can  be 
moistened  by  water,  or  two  small  balls  of  wax,  which  cannot 
be  moistened  by  water,  be  placed  in  a  vessel  of  water,  and 
brought  so  near  each  other  that  the  surfaces  of  capillary 
elevation  or  depression  interfere,  the  balls  will  attract  each 
other  and  come  together.  If  one  ball  of  wood  and  one  of 
wax  be  brought  so  near  that  the  surfaces  of  capillary  eleva- 
tion and  depression  interfere,  the  bodies  will  repel  each 
other  and  separate.  If  two  needles  be  carefully  oiled  and 
laid  upon  the  surface  of  a  vessel  of  water,  they  will  repel 
the  water  from  their  neighborhood,  and  float.  If,  whilst 
floating,  they  are  brought  sufficiently  near  to  each  other  to 
permit  the  surfaces  of  capillary  depression  to  interfere,  the 
needles  will  immediately  rush  together.  The  reason  of  the 
needles  floating  is,  that  they  repel  the  water,  heaping  it  up 
on  each  side,  thus  forming  a  cavity  in  the  surface  ;  the 
needle  is  buoyed  up  by  a  force  equal  to  the  weight  of  the 
displaced  fluid,  and,  when  this  exceeds  the  weight  of  the 


MECHANICS    OF    LIQUID8.  283 

ueedle,  it  will  float.  It  is  on  this  principle  that  certain 
insects  move  freely  over  the  surface  of  a  sheet  of  water; 
their  feet  are  lubricated  with  an  oily  substance  which  repels 
the  water  from  around  them,  producing  a  hollow  around 
each  foot,  and  giving  rise  to  a  buoyant  effort  greater  than 
the  weight  of  the  insect. 

The  principle  of  mutual  attraction  between  bodies,  both 
of  which  repel  water,  or  both  of  which  attract  it,  accounts 
for  the  fact  that  small  floating  bodies  have  a  tendency  to 
collect  in  groups  about  the  borders  of  the  containing  vessel. 
When  the  material  of  which  the  vessel  is  made,  exercises  a 
different  capillary  action  from  that  of  the  floating  particles, 
they  will  aggregate  themselves  at  a  distance  from  the  sur- 
face of  the  vessel. 

Applications  of  the  Principles  of  Capillarity. 

184.  It  is  in  consequence  of  capillary  action  that  water 
rises  to  fill  the  pores  of  a  sponge,  or  of  a  lump  of  sugar. 
The  same  principle,  causes  the  oil  to  rise  in  the  wick  of  a 
lamp,  which  is  but  a  bundle  of  fibres  very  nearly  in  contact, 
leaving  capillary  interstices  between  them. 

The  siphon  filter  differs  but  little  in  principle  from  the 
wick  of  a  lamp.  It  consists  of  a  bundle  of  fibres  like  a 
lamp-wick,  one  end  of  which  dips  into  a  vessel  of  the  liquid 
to  be  filtered,  whilst  the  other  hangs  over  the  edge  of  the 
vessel.  The  liquid  ascends  the  fibrous  mass  by  the  principle 
of  capillary  attraction,  and  continues  to  advance  till  it 
reaches  the  overhanging  end,  when,  if  this  is  lower  than  the 
upper  surface  ot  the  liquid,  the  liquid  will  fall  by  drops  from 
the  end  of  the  wick,  the  impurities  being  left  behind. 

The  principle  of  capillary  attraction  is  used  for  splitting 
rocks  and  raising  weights.  To  employ  this  principle  in 
cleaving  mill-stones,  as  is  done  in  France,  the  stone  is  first 
dressed  to  the  form  of  a  cylinder  of  the  required  diameter 
for  the  mill-stone.  Grooves  are  then  cut  a-ound  it  where 
the  divisions  are  to  take  place,  and  into  these  grooves 
thoroughly  dried  wedges  of  willow-wood  are  driven.  On 
being  exposed  to  the  action  of  moisture,  the  cells  of  the 


IT  S4:  MECHANICS. 

wood  absorb  a  large  quantity  of  water,  expand,  and  finally 
split  the  rock. 

To  raise  a  weight,  let  a  thoroughly  dry  cord  be  fastened 
to  the  weight,  and  then  stretched  to  a  point  above.  If,  now, 
the  cord  be  moistened,  the  fibres  will  absorb  the  moisture, 
expanding  laterally,  the  rope  will  be  diminished  in  length, 
and  the  weight  raised. 

The  principle  of  capillary  attraction  is  also  very  exten- 
sively employed  in  metallurgy,  in  a  process  of  purifying 
metals,  called  cupellation. 

Endosmose  and  Exosmose. 

185.  The  names  endosmose  and  exosmose  have  been 
given  to  two  currents  flowing  in  a  contrary  direction 
between  two  liquids,  when  they  are  separated  by  a  thin 
porous  partition,  either  organic  or  inorganic.  The  discovery 
of  this  phenomena  is  due  to  M.  Dutrochet,  who  called  the 
flowing  in,  endosmose,  and  the  flowing  out,  exosmose.  The 
existence  of  the  currents  was  established  by  means  of  an 
instrument,  to  which  he  gave  the  name  endosmometre.  This 
instrument  consists  of  a  long  tube  of  glass,  at  one  end  of 
which  is  attached  a  membranous  sack,  secured  by  a  tight 
ligature.  If  the  sack  is  filled  with  gum  water,  a  solution  of 
sugar,  albumen,  or,  in  fact,  with  almost  any  solution  denser 
than  water,  and  then  plunged  into  water,  it  is  observed, 
after  a  time,  that  the  fluid  rises  in  the  stem,  and  is  depressed 
in  the  vessel,  showing  that  water  has  entered  the  sack  by- 
passing through  the  pores.  By  applying  suitable  tests,  it  is 
also  found,  that  a  portion  cf  the  liquid  in  the  sack  has  passed 
through  the  pores  into  the  vessel. 

Two  currents  are  thus  established.  If  the  operation 
be  reversed,  and  the  bladder  and  tube  be  filled  with  pure 
water,  the  liquid  in  the  vessel  will  rise,  whilst  that  in  the 
tube  falls.  The  phenomena  of  endosmose  and  exosmose 
are  extremely  various,  and  serve  to  explain  a  great  variety 
of  interesting  facts  in  animal  and  vegetable  physiology. 
The  cause  of  the  currents  is  the  action  of  molecular  forces 
exerted  between  the  particles  of  the  bodies  employed. 


MECHANICS    OF    GASES    AND    VAPORS.  285 


CHAPTER   VIII. 

MECHANIC?       OF      GASES      AND      VAPORS, 

Gases  and  Vapors. 

186.  Gases  and  vapors  are  distinguished  from  other 
fluids,  by  their  great  compressibility,  and  correspondingly 
great  elasticity.  These  fluids  continually  tend  to  occupy  a 
greater  space  ;  this  expansion  goes  on  till  counteracted  by 
some  extraneous  force,  as  that  of  gravity,  or  the  resistance 
offered  by  a  containing  vessel. 

The  force  of  expansion,  which  is  common  to  all  gases  and 
vapors,  is  called  their  tension  or  elastic  force.  We  shall 
take  for  the  unit  of  this  force  at  any  point,  the  pressure 
which  would  be  exerted  upon  a  square  inch  of  surface,  were 
the  pressure  the  same  at  every  point  of  the  square  inch  as 
at  the  point  in  question.  If  we  denote  this  unit,  by  p,  the 
area  pressed,  by  a,  and  the  entire  pressure,  by  P,  we  shall 
have, 

P  =  ap (151.) 

Most  of  the  principles  already  demonstrated  for  liquids 
hold  good  for  gases  and  vapors,  but  there  are  certain  pro- 
perties arising  from  elasticity  which  are  peculiar  to  aeriform 
fluids,  some  of  which  it  is  now  proposed  to  investigate. 

Atmospheric  Air. 

187.  The  gaseous  fluid  which  envelops  our  globe,  and 
extends  on  all  sides  to  a  distance  of  many  miles,  is  called  the 
atmosphere.  It  consists  principally  of  nitrogen  and  oxygen, 
together  with  variable,  but  small  portions  of  watery  vapor 
and  carbonic  acid,  all  in  a  state  of  mixture.  On  an  average, 
it  is  found  by  experiment  that   1000  parts  by  volume  of 


286  MECHANICS. 

atmospheric  air,  taken  near  the  surface  of  the  earth,  coi  slsta 
of  about, 

788  parts  of  nitrogen, 
197  parts  of  oxygen, 
14  parts  of  watery  vapor, 
1  part  of  carbonic  acid. 

The  atmosphere  may,  physically  speaking,  be  taken  as  a 
type  of  gases,  for  it  is  found  by  experiment  that  the  laws 
regulating  the  density,  expansibility,  and  elasticity,  are  the 
same  for  all  gases  and  vapors,  so  long  as  they  maintain  a 
purely  gaseous  form.  It  is  found,  however,  in  the  case  of 
vapors,  and  of  those  gases  which  have  been  reduced  to  a 
liquid  form,  that  the  law  changes  just  before  actual  lique- 
faction. 

This  change  appears  to  be  somewhat  analagous  to  that 
observed  when  water  passes  from  the  liquid  to  the  solid 
form.  Although  water  does  not  actually  freeze  till  reduced 
to  a  temperature  of  32°  Fah.,  it  is  found  that  it  reaches  its 
maximum  density  at  about  38°. 75,  at  which  temperature  the 
particles  seem  to  commence  arranging  themselves  according 
to  some  new  laws,  preparatory  to  taking  the  solid  form. 
Atmospheric  Pressure. 

188.     If  a  tube,  35  or  36  inches  long,  open  at  one  end 
and  closed  at  the  other,  be  filled  with  pure  mercury,  and 
inverted  in  a  basin  of  the  same,  it  is  observed 
that  the  mercury  will  fall  in  the  tube  until  the  C 

vertical  distance  from  the  surface  of  the  mer-  b 

cury  in  the  tube  to  that  in  the  basin  is  about  30 
inches.     This  column  of  mercury  is  sustained  by 
the  pressure  of  the  atmosphere    exerted   upon 
the   surface  of  the   mercury  in  the   basin,  and 
transmitted  through  the  fluid,  according  to  the 
general  law  of  transmission  of  pressures.     The         ApH 
column  of  mercury  sustained  by  the  elasticity  ot         ~    162> 
the  atmosphere  is  called  the  barometric  column, 
because  it  is  generally  measured  by  an  instrument  called  a 
barometer.     In  fact,  the  instrument  just  described,  when 


MECHANICS    OF    GASES    AND    VAPORS.  287 

provided  with  a  suit  able  scale  for  measuring  the  altitude  of 
the  column,  is  1  complete  barometer.  The  height  of  the 
barometric  column  fluctuates  somewhat,  even  at  the  same 
place,  on  account  of  changes  of  temperature,  and  other 
causes  yet  to  be  considered. 

Observation  has  shown,  that  the  average  height  of  the 
barometric  column  at  the  level  of  the  sea,  is  a  trifle  less  than 
30  inches. 

The  weight  of  a  column  of  mercury  30  inches  in  height, 
having  a  cross  section  of  one  square  inch,  is  nearly  15 
pounds.  Hence,  the  unit  of  atmospheric  pressure  at  the 
level  of  the  sea,  is  15  pounds. 

This  unit  is  called  an  atmosphere,  and  is  often  employed 
in  estimating  the  pressure  of  elastic  fluids,  particularly  in 
the  case  of  steam.  Hence,  to  say  that  the  pressure  of  steam 
in  a  boiler  is  two  atmospheres,  is  equivalent  to  saying,  that 
there  is  a  pressure  of  30  pounds  upon  each  square  inch  of 
the  interior  of  the  boiler.  In  general,  when  we  say  that  the 
tension  of  a  gas  or  vapor  is  n  atmospheres,  we  mean  that 
each  square  inch  is  pressed  by  a  force  of  n  times  15  pounds. 

Mariotte's  Law. 

189.  When  a  given  mass  of  any  gas  or  vapor  is  com- 
pressed so  as  to  occupy  a  smaller  space,  other  things  being- 
equal,  its  elastic  force  is  increased  ;  on  the  contrary,  if  its 
volume  is  increased,  its  elastic  force  is  diminished. 

The  law  of  increase  and  diminution  of  elastic  force,  first 
discovered  by  Mariotte,  and  bearing  his  name,  may  be 
enunciated  as  follows : 

The  elastic  force  of  a  given  mass  of  any  gas,  ichose  tem- 
perature 'remains  the  same,  varies  inversely  as  the  volume 
which  it  occupies. 

,  As  long  as  the  mass  remains  the  same,  the  density  must 
vary  inversely  as  the  volume  occupied.  Hence,  from  Mari- 
otte's Law,  it  follows,  that, 

The  elastic  force  of  any  gas,  whose  temperature  remains 
the  same,  varies  as  its  density,  and  conversely,  the  density 
varies  as  the  elastic  force. 


ITTd 

p    P 

K       li 

■       c 

2S8  MECHANICS. 

Mariotte's  law  may  be  verified  in  the  case  of  atmosplierio 
air,  by  the  aid  of  an  instrument  called  Mamotte's  Tube. 
This  instrument  consists  of  a  tube  AH  CD,  of  uniform  bore* 
bent  so  that  its  two  branches  are  parallel  to  each 
other.     The  shorter  branch  AP,  is  closed  at  its 
upper  extremity,  whilst  the  longer  one  remains 
open  for  the  reception  of  mercury.     Between  the 
two  branches  of  the  tube,  and  attached  to  the 
same  frame  with  it,  is  a  scale  of  equal  parts  for 
measuring  distances. 

To  use  the  instrument,  place  it  in  a  vertical 
position,  and  pour  mercury  into  the  tube,  until  it 
just  cuts  off  the  communication  between  the  two 
branches  The  mercury  will  then  stand  at  the 
same  level  PC,  in  both  branches,  and  the  tension 
of  the  confined  air  in  AB,  will  be  exactly  equal  to  that  of 
the  external  atmosphere.  If  an  additional  quantity  of  mer- 
cury be  poured  into  the  longer  branch,  the  confined  air  in 
the  shorter  branch  will  be  compressed,  and  the  mercury 
will  rise  in  both  branches,  but  higher  in  the  longer,  than  in 
the  shorter  one.  Suppose  the  mercury  to  have  risen  in  the 
shorter  branch,  to  K,  and  in  the  longer  one,  to  P.  There 
will  be  an  equilibrium  in  the  mercury  lying  below  the  hori- 
zontal plane  KK;  there  will  also  be  an  equilibrium  between 
the  tension  of  the  air  in  AK,  and  the  forces  which  give  rise 
to  that  tension.  These  forces  are  the  pressure  of  the  exter- 
nal atmosphere  transmitted  through  the  mercury,  and  the 
weight  of  a  column  of  mercury  whose  base  is  the  cross-sec- 
tion of  the  tube,  and  whose  altitude  is  PK.  If  we  denote 
the  height  of  the  column  of  mercury  which  will  be  sustained 
by  the  pressure  of  the  external  atmosphere,  by  h,  the  ten- 
sion of  the  air  in  AK,  will  be  measured  by  the  weight  of  a 
column  of  mercury,  whose  base  is  the  cross-section  of  the* 
tube,  an3  whose  height  is  li  +  PK.  Since  the  weight  is 
proportional  to  the  height,  the  tension  of  the  confined  air 
will  be  proportional  to   h  -f-  PK. 

Now,  whatever  may  be  the  value  of  PK,  it  is  found  that, 


MECHANICS    OF    GASES    AND    VAPORS.  289 

AB  .  h 


AK  = 


h  +  PK 


If  PK  =  h,  we  shall  have,  AK '=  1.4J5;  if  P/iT=  2A, 
we  shall  have,  ^4A"  =  %AB ;  in  general,  if  PK  =  nA,  w 
being  any  positive  number,  either  entire  or  fractional,  we 

AB 

shall  have,  AK  —  •     Mariotte's  Law  was  verified 

in  this  manner  by  Dulong  and  Arago  for  all  values  of  w,  up 
to  n  —  27.  The  law  may  also  be  verified  when  the  pres- 
sure is  less  than  an  atmosphere,  by  means  of  the  following 
apparatus.    - 

AK  represents  a  straight  tube  of  uniform  bore,  closed  at 
its  upper  and  open  at  its  lower  extremity :  CD 
is  a  long  cistern  of  mercury.  The  tube  AK  is 
either  graduated  into  equal  parts,  commencing 
at  A,  or  it  has  attached  to  it  a  scale  of  brass  or 
ivory. 

To  use  the  instrument,  pour  mercury  into  the 
tube  till  it  is  nearly  full ;  place  the  finger  over 
the  open  end,  and  invert  it  in  the  cistern  of  mer- 
cury, and  depress  it  till  the  mercury  stands  at 
the  same  level  without,  as  within  the  tube,  and 
suppose  the  surface  of  the  mercury  in  this  case  Fig;  164 

to  cut  the  tube  at  B.  Then  will  the  tension 
of  the  confined  air  in  AB,  be  equal  to  that  of  the  external 
atmosphere.  If  now  the  tube  be  raised  vertically,  the  air  in 
AB  will  expand,  its  tension  will  diminish  and  the  mercury 
will  fall  in  the  tube,  to  maintain  the  equlibrium.  Suppose 
the  level  of  the  mercury  in  the  tube  to  have  reached 
the  point  K  In  this  position  of  the  instrument  the  tension 
of  the  air  in  AK,  added  to  the  weight  of  the  column  of  mer- 
cury, KE  will  be  equal  to  the  tension  of  the  external  air. 

Now,  it  is  found,  whatever  may  be  the  value  of  KE,  that 


r,A 

LB 


h-EK 
13 


290  MECHANICS. 

If  EK  =  iA,  we  have,  AK  =  2AB;  if  EK :,  §A,  we 

have,  w4/r  =  3^4^;  in  general,  if  EK  =  A,  we  have, 

ATI  ft  +  1    ' 

n  +  1 

aIakiotte's  law  has  been  verified  in  this  manner,  for  all 
values  of  n,  up  to  n  —  111. 

It  is  a  law  of  Physics  that,  when  a  gas  is  suddenly  com- 
pressed, heat  is  evolved,  and  when  a  gas  is  suddenly  ex- 
panded, heat  is  absorbed  ;  hence,  in  making  the  experiment, 
care  must  be  taken  to  have  the  temperature  kept  uniform. 

Gay  Lussac's  Law. 

190.  If,  whilst  the  volume  of  any  gas  or  vapor  remains 
the  same,  its  temperature  be  increased,  its  tension  is  in- 
creased also.  If  the  pressure  remain  the  same,  the  volume 
of  the  gas  will  increase  as  the  temperature  is  raised.  The 
law  of  increase  and  diminution,  as  deduced  by  Gay  Lussac, 
whose  name  it  bears,  may  be  enunciated  as  follows : 

In  a  given  mass  of  any  gas,  or  vapor,  if  the  volume 
remains  the  same,  the  tension  varies  as  the  temperature  /  if 
the  tension  remains  the  same,  the  volume  varies  as  the  tem- 
perature. 

According  to  Regnault,  if  a  given  mass  of  atmospheric 
air  be  heated  from  32°  Fahrenheit  to  212°,  the  tension,  or 
pressure  remaining  constant,  its  volume  will  be  increased  by 
the  .3G65th  part  of  the  volume  at  32°.  Hence,  the  increase 
of  volume  for  each  degree  of  temperature  is  the  .00204th  part 
of  the  volume  at  32°.  If  we  denote  the  volume  at  32°  by  v, 
and  the  volume  at  the  temperature  t\  by  v',  we  si  all  there- 
fore have, 

v'  =  v[l  +  .00204(2'-  32)]     .     .     (  152.) 

Solving  with  reference  to  v,  we  have, 

v' 


"  1  +  .00204(2'-  32)  (lo3.) 

Formula  (153)  enables  us  to  compute  the  volume  of  any 


MECHANICS    OF    GASES    AND    VAPOWS. 


291 


mass  of  air  at  32°,  knowing  its  volume  at  the  temperature 
t\  the  pressure  remaining  constant. 

To  find  the  volume  at  the  temperature  t",  we  have  simply 
to  substitute  t"  for  t'  in  (152.)  Denoting  this  volume  by 
v'\  we  have, 

V"=  v[l  +  .00204(£"  —  32)]. 


Substituting  for  v  its  value  from  (153),  we  get, 

,1  4-  .00204(r—  32) 
1  +  .00204(£'  -  32) 


(154.) 


This  formula  enables  us  to  compute  the  volume  of  any 
mass  of  air,  at  a  temperature  t",  when  we  know  its  volume 
at  the  temperature  t' ;  and,  since  the  density  varies  in- 
versely as  the  volume,  we  may  also,  by  means  of  the  same 
formula,  find  the  density  of  any  mass  of  air,  at  the  temper- 
ature t'\  when  we  have  given  its  density  at  the  tempera- 
ture t\ 

Manometers. 

191.  A  manometer  is  an  instrument  used  for  measuring 
the  tension  of  gases  and  vapors,  and  particularly  of  steam. 
Two  principle  varieties  of  manometers  are  used  for  measur- 
ing the  tension  of  steam,  the  open  manometer,  and  the 
dosed  manometer. 

The  open  Manometer. 

192.  The  open  manometer  consists,  essentially,  of  an 
open  glass  tube  A  J?,  terminating  below, 

nearly  at  the  bottom  of  a  cistern  EF. 
The  cistern  is  of  wrought  iron,  steam 
tight,  and  filled  with  mercury.  Its  dimen- 
sions are  such,  that  the  upper  surface  of 
the  mercury  will  not  be  materially  lowered, 
when  a  portion  of  the  mercury  is  forced 
up  the  tube.  ED  is  a  tube,  by  means  of 
which,  steam  may  be  admitted  from  the 
boiler  to  the  surface  of  the  mercury  in  the 
cistern.     This  tube  is  sometimes  filled  with  Fig.  165. 


lb 


292  MECHANICS. 

water,  through  which  the  pressure  of  the  steam  is  trans 
mitted  to  the  mercury. 

To  graduate  the  instrument.  All  communication  with 
the  boiler  is  cut  off",  by  closing  the  stop-cock  E,  and  commu- 
nication with  the  external  air  is  made  by  opening  the  stop- 
cock I).  The  point  of  the  tube  AB,  to  which  the  mercury 
rises,  is  noted,  and  a  distance  is  laid  off",  upwards,  from  this 
point,  equal  to  what  the  barometric  column  wants  of  30 
inches,  and  the  point  .//"thus  determined,  is  marked  1.  This 
point  will  be  very  near  the  surface  of  the  mercury  in  the 
cistern.  From  the  point  II,  distances  of  30,  60,  90,  &c, 
inches  are  laid  off  upwards,  and  the  corresponding  points 
numbered  2,  3,  4,  &c.  These  divisions  correspond  to 
atmospheres,  and  may  be  subdivided  into  tenths  and 
hundredths. 

To  use  the  instrument,  the  stop-cock  D  is  closed,  and  a 
communication  made  with  the  boiler,  by  opening  the  stop- 
cock E.  The  height  to  which  the  mercury  rises  in  the 
tube,  will  indicate  the  tension  of  the  steam  in  the  boiler, 
which  may  be  read  from  the  scale  in  terms  of  atmospheres 
and  decimals  of  an  atmosphere.  If  the  pressure  in  pounds 
is  wished,  it  may  at  once  be  found,  by  multiplying  the 
reading  of  the  instrument  by  15. 

The  principal  objection  to  this  kind  of  manometer,  is  its 
want  of  portability,  and  the  great  length  of  tube  required, 
when  high  tensions  are  to  be  measured. 

The  closed  Manometer. 

193.  The  general  construction  of  the  closed  manometer 
is  the  same  as  that  of  the  open  manometer,  with  the  excep- 
tion that  the  tube  AB  is  closed  at  the  top.  The  air  which 
is  confined  in  the  tube,  is  then  compressed  in  the  same  way 
as  in  Makiotte's  tube. 

To  graduate  this  instrument,  We  determine  the  division 
II,  as  before.  The  remaining  divisions  are  found  by  apply- 
ing Mariotte's  law. 

Denote  the  distance  in  inches,  from  II  to  the  top  of  the 


MECHANICS    OF   GASES    AND   VAPORS.  293 

tube,  by  I;  the  pressure  on  the  mercury,  expressed  in 
atmospheres,  by  ?i,  and  the  distance  in  inches,  from  II  to  the 
upper  surface  of  the  mercury  in  the  tube,  by  x. 

The  tension  of  the  air  in  the  tube  will  be  equal  to  that  on 
the  mercury  in  the  cistern,  diminished  by  the  weight  of  a 
column  of  mercury,  whose  altitude  is  x.  Hence,  in  atmos- 
pheres, it  is 

x 

The  bore  of  the  tube  being  uniform,  the  volume  occupied 

by  the  compressed  air  will  be  proportional  to  its  height. 

When  the  pressure  is  1  atmosphere,  the  height  is  £;  when 

x 
the  pressure  is  n atmospheres,   the  height   is  I  —  x. 

Hence,  from  Mariotte's  law, 

X  7  7 

1  :  n  —  —  :  :  I  —  x  :  I . 
30 

Whence,  by  reduction, 

x>  _  (30rc  H-  l)x  =  —  30l(n  —  1). 
Solving,  wTith  respect  to  cc,  wre  have, 

SOn  +  1    ,       /     ~T,        ~    ,    /30^  +  A2 


x  — 


The  upper  sign  of  the  radical  is  not  used,  as  it  would  give 
a  value  for  x,  greater  than  /.  Taking  the  lower  sign,  and,  a? 
a  particular  case,  assuming  I  —  30  in.,  we  have, 


x  -  15^  +-15  —  y/  —  900(n  —  1)  +  {Ion  +  15)2. 

Making  n  =  2,  3,  4,  &c,  in  succession,  we  find  for  x,  the 
corresponding  values,  11.46  in.,  17.58  in.,  20.92  in.,  &c. 
These  distances  being  set  off  from  II,  upwards,  and  marked 
2,  3,  4,  <fcc,  indicate  atmospheres.  The  intermediate  spaces 
are  subdivided  by  means  of  the  same  formula. 


294 


MECHANICS. 


B 


Fig.  166. 


The  use  of  this  instrument  is  the  same  as  that  of  the 
manometer  last  described. 

In  making  the  graduation,  we  have  supposed  the  tem- 
perature to  remain  the  same.  If,  however,  it  does  not 
remain  the  same,  the  reading  of  the  instrument  must  be 
corrected  by  means?  of  a  table  computed  for  the  \  urpose. 

The  instruments  already  described,  can  only  be  used  for 
measuring  tensions  greater  than  one  atmosphere. 

The  Siphon  Guage. 

194.  The  siphon  guage  is  an  instrument  employed  to 
measure   tensions   of  gases    and  vapors, 

when  they  are  less  than  an  atmosphere. 
It  consists  of  a  tube  AB  C\  bent  so  that 
its  two  branches  are  parallel.  The  branch 
BC  is  closed  at  the  top,  and  filled  with 
mercury,  which  is  retained  by  the  pres- 
sure of  the  atmosphere,  whilst  the  branch 
AB  is  open  at  the  top.  If,  now,  the  air 
be  rarified  in  any  manner,  or  if  the  mouth 
A  of  the  tube,  be  exposed  to  the  action  of  any  gas  whose 
tension  is  sufficiently  small,  the  mercury  will  no  longer  be 
supported  in  the  branch  BC,  but  will  fall  in  that  and  rise  in 
the  other.  The  distance  between  the  surfaces  of  the  nier 
cury  in  the  two  branches,  as  given  by  a  scale  placed  between 
them,  will  indicate  the  tension  of  the  gas.  If  this  distance 
is  expressed  in  inches,  the  tension  can  be  found,  in  atmos- 
pheres, by  dividing  by  30,  or,  in  pounds,  by  dividing  by  2. 
The  Diving-Bell. 

195.  The  diving-bell  is  a  bell-shaped  vessel,  open  at 
the  bottom,  used  for  descending  below  the 

surface  of  the  water.  The  bell  is  placed 
bo  that  its  mouth  shall  continue  horizontal, 
and  is  let  down  by  means  of  a  rope  AB, 
and  the  whole  apparatus  is  sunk  by 
weights  properly  adjusted.  The  air  con- 
tained in  the  bell  before  immersion,  will 
be    compressed   by   the   weight    of   the 


\yJ 


Fig.  167. 


MECHANICS    OF   GASES    AND   VAPOR8.  205 

water,  but  its  increased  elasticity  will  prevent  the  water 
from  rising  to  the  top  of  the  bell,  which  is  provided  with 
seats  for  the  accommodation  of  those  wishing  to  descend. 
The  air  within  is  constantly  contaminated  by  breathing,  and 
is  continually  replaced  by  fresh  air,  pump  id  in  through  a 
tube  EG.  Were  there  no  additional  air  introduced,  the 
volume  of  the  compressed  air,  at  any  depth,  might  be  com- 
puted by  Mariotte's  law.  The  unit  of  the  compressing 
force,  in  this  case,  is  the  weight  of  a  column  of  water  whose 
cross-section  is  a  square  inch,  and  whose  height  is  the 
distance  from  DC,  to  the  surface  of  the  water. 
The  Barometer. 

196.  The  barometer  is  an  instrument  for  measuring 
the  pressure  of  the  atmosphere.  As  already  explained,  it 
consists  of  a  glass  tube,  hermetically  sealed  at  one  extre- 
mity, which  is  filled  writh  mercury,  and  inverted  in  a  basin 
of  that  fluid.  The  pressure  of  the  air  is  indicated  by  the 
height  of  the  column  of  mercury  which  it  supports. 

A  great  variety  of  forms  of  the  mercurial  barometer  have 

been  devised,  all  invoking  the  same  mechanical  principle. 

The  two  most  important  of  these  are  the  siphon  and  the 

cistern  barometer. 

The  Siphon  Barometer. 

197.  The  siphon  barometer  consists  essentially  of  a 
tube  CDE,  bent  so  that  its  two  branches,  CD 

and  DE,  shall  be  parallel  to  each  other.      A  -c 

scale  of  equal  parts  is  placed  between  them,  i\\x 

and  attached  to  the  same  frame  with  the  tube. 
The  longer  branch  CD,  is  about  32  or  33 
inches  in  length,  hermetically  sealed  at  the  top, 
and  filled  with  mercury ;  the  shorter  one  is 
open  to  the  action  of  the  air.  When  the 
instrument  is  placed  vertically,  the  mercury 
sinks  in   the   longer   branch   and   rises  in  th^  B 

shorter  one.      The  distance  between  the  sur-        Fls'  m 
face  of  the  mercury  in  the  two  branches,  as  measured  by 
the  scale  of  equal  parts,  indicates  the  pressure  of  the  atniofr 
phere  at  the  particular  time  and  place. 


I 


296 


MECHANICS. 


K 


The  Cistern  Barometer. 

198.  The  cistern  barometer  consists  of  a  glass  tube, 
filled  and  inverted  in  a  cistern  of  mercury,  as  already 
explained.  The  tube  is  surrounded  by  a  frame  of  metal, 
firmly  attached  to  the  cistern.  Two  opposite  longitudinal 
openings,  near  the  upper  part  of  the  frame,  permit  the 
upper  surface  of  the  mercury  to  be  seen.  A  slide,  moved 
up  and  down  by  means  of  a  rack  and  pinion,  may  be 
brought  exactly  to  the  upper  level  of  the  mercury.  The 
height  of  the  column  is  then  read  from  a  scale,  so  adjusted  as 
to  have  its  0  at  the  surface  of  the  mercury  in  the  cistern. 
The  scale  is  graduated  to  inches  and  tenths,  and  the  smaller 
divisions  are  read  by  means  of  a  vernier. 

The  figure  shows  the  arrangement  of  parts  in 
a  complete  cistern  barometer.  KK  represents 
the  frame  of  the  barometer;  IIII  that  of  the 
cistern,  open  at  the  upper  part,  that  the  level 
of  the  mercury  in  the  cistern  may  be  seen 
through  the  glass;  X,  an  attached  thermo- 
meter, to  show  the  temperature  of  the  mer- 
cury in  the  tube ;  iV,  a  part  of  the  sliding  ring 
bearing  the  vernier,  and  moved  up  and  down 
by  the  milled-headed  screw  M. 

The  particular  arrangement  of  the  cistern  is 
shown  on  an  enlarged  scale  in  Fig.  ]  70.  A 
represents  the  barometer  tube,  terminating  in 
a  small  opening,  to  prevent  too  sudden  shocks 
when  the  instrument  is  moved  from  place  to 
place  ;  11  represents  the  frame  of  the  cis- 
tern; 7>,  the  upper  portion  of  the  cistern, 
made  of  glass,  that  the  surface  of  the  mercury 
may  be  seen;  Tv,  a  conical  piece  of  ivory,  pro- 
jecting  from  the  upper  surface  of  the  cistern  : 
when  the  surface  of  the  mercury  just  touches 
the  point  of  the  ivory,  it  is  -it  the  0  of  the 
scale;  CC  represents  the  Lower  part  of  the 
cifftern,  and  is  made  of  leather,  or  some  other  Fi    17(| 


•p~T;  rrk 

Q1[ 


J) 
Fig.  169. 


r 

A 

15 

\ 

a 

I 

TT 

MECHANICS    OF   GASES   AND   VAPORS.  297 

flexible  substance,  and  firmly  attached  to  the  glass  part ; 
J)  is  a  screw,  working  through  the  bottom  of  the  frame,  and 
against  the  bottom  of  the  bag  (7(7,  through  the  medium  of 
a  plate  P.  The  screw  D,  serves  to  bring  the  surface  of  the 
mercury  to  the  point  of  the  ivory  piece  E,  and  also  to  force 
the  mercury  up  to  the  top  of  the  tube,  when  it  is  desired 
to  transport  the  barometer  from  place  to  place. 

To  use  this  barometer,  it  should  be  suspended  vertically, 
and  the  level  of  the  mercury  in  the  cistern  brought  to  the 
point  of  the  ivory  piece  E,  by  means  of  the  screw  D ;  \  a 
smart  rap  with  a  key  upon  the  frame  will  detach  the  mer- 
cury from  the  glass  to  which  it  sometimes  tends  to  adhere. 
The  sliding  ring  jV,  is  next  run  up  or  down  by  means  of  the 
screw  J/,  till  its  lower  edge  appears  tangent  to  the  upper 
surface  of  the  mercury  in  the  tube,  and  the  altitude  is  read 
from  the  scale.  The  height  of  the  attached  thermometer 
should  also  be  noted. 

The  requirements  of  a  good  barometer  are,  sufficient 
width  of  tube,  perfect  purity  of  the  mercury,  and  a  scale 
with  a  vernier  accurately  graduated  and  adjusted. 

The  bore  of  the  tube  should  be  as  large  as  practicable,  to 
diminish  the  effect  of  capillary  action.  On  account  of  the 
mutual  repulsion  between  the  particles  of  the  glass  and  mer- 
cury, the  mercury  is  depressed  in  the  tube,  and  this  depres- 
sion increases  as  the  diameter  of  the  tube  diminishes. 

In  all  cases,  this  depression  should  be  allowed  for,  and 
corrected  by  means  of  a  table  computed  for  the  purpose. 

To  secure  purity  of  the  mercury,  it  should  be  carefully 
distilled,  and  after  the  tube  is  filled,  it  should  be  boiled  over 
a  spirit-lamp,  to  drive  off  any  bubbles  of  air  that  might  ad- 
here to  the  walls  of  the  tube. 

Uses  of  the  Barometer. 

199.  The  primary  object  of  the  barometer  is,  to  meas- 
ure the  pressure  of  the  atmosphere  at  any  time  or  place.  It 
is  used  by  mariners  and  others,  as  a  weather-glass.  It  is 
also  extensively  employed  for  determining  the  heights  of 
points  on  the  earth's  surface,  above  the  level  of  the  ocean. 
13* 


298  MECHANICS. 

The  principle  on  which  it  is  employed  for  the  latter  pur 
pose  is,  that  the  pressure  of  the  atmosphere  at  any  place 
depends  upon  the  weight  of  a  column  of  air  reaching  from 
the  place  to  the  upper  limit  of  the  atmosphere.  As  we  as- 
cend above  the  level  of  the  ocean,  the  weight  of  the  column 
diminishes ;  consequently,  the  pressure  becomes  less,  a  tact 
which  is  shown  by  the  mercury  falling  in  the  tube.  We 
shall  investigate  a  formula  for  determining  the  difference  of 
level  between  any  two  points. 

Difference  of  Level. 

200.  Let  aB  represent  a  portion  of  a  vertical  prism  of 
air,  whose  cross-section  is  one  square  inch.  De- 
note the  pressure  on  the  lower  base  .2?,  by  p,  and 
on  the  upper  base  aa\  by  p' ;  denote  the  density 
of  the  air  at  JBy  by  <7,  and  at  aa\  by  d\  and  sup- 
pose the  temperature  throughout  the  column  to 
be  32°  Fah. 

Pass  a  horizontal  plane  bb\  infinitely  near  to 
(/'/',  and   denote  the  weight   of  the  elementary        F1    m 
volume  of  air  ob,  by  to.      Conceive   the  entire 
column  to  be  divided  by  horizontal  planes  into  elementary 
prisms,  such  that  the  weights  of  each  shall  be  equal  to  w, 
and  denote  their  heights,  beginning  at  «,  by  s,  s\  s",  &c. 

From  Makiotte's  law,  we  shall  have, 

y     _  <? 

p         d 

The  air  throughout  each  elementary  prism  may  be  re- 
garded as  homogeneous ;  hence,  the  density  of  the  air  in  db 
is  equal  to  its  weight,  divided  by  its  volume  into  gravity 
(Art.  12).     But  its  volume  is  equal  to    1   x  1   x  s  =  s 
hence, 

#  =  ?.■ 

g* 

Substituting  this  in  the  preceding  equation,  we  have, 


whence, 


MECHANICS   OF   GASES    AND   VAPOR8.  299 

P'  W 

p  ~~  gsd' 


2-  X»,     •     •     •     .     (155.] 

dg      p'  v 


From  Davies'  Bourdon,  page  297,  we  have,  by  substitute 

w 
ing  for  y  the  fraction  —  ,  the  equation, 

»/  10  \  10  10*  U0%  0 

<1+j?)  =?-**  + **-*■■ 

10 

But  —  being  infinitely  small,  all  the  terms  in  the  second 
member,  after  the  first,  may  be  neglected,  giving, 

10  ./,      .     W\  10  _/»'  +  W\ 

p  =  V+p)'  or>  f  =  z(V_); 

or  finally, 

|  =  l(p>  +  w)  -  lp\ 

in  which  I  denotes  the  Napierian  logarithm. 

In  this  equation,  p'  denotes  the  pressure  on  the  prism  ab ; 

hence,  p'  -f  uo    denotes   the   pressure    on   the   next   prism 

below,  that  is,  on  the  prism  be. 

w 
If  we  substitute  this  value  of  —   in  Equation  (155),  we 

shall  have,  for  the  height  of  the  prism  ab, 

Substituting  in  succession  for  p',  the  values  p'-\-  w,p'  +  2w>, 
p'  +  3w,  &c,  we  shall  find  the  heights  of  the  elementary 
prisms  be,  ed,  &v..     Wo  shall  therefore  have, 


300  MECHANICS. 


8  =  !k[l{pf+  ^-^ 


*n'=  J^P(*'+  nw?)  -  ?0p'  +  (;i  -  WJ- 

If  w  denote  the  number  of  elementary  prisms  in  AB,  the 
sum  of  the  first  members  will  be  equal  to  AB.  Adding  the 
equations  member  to  member,  and  denoting  the  sum  of  the 
first  members  by  2,  we  have, 

Because  nw  denotes  the  weight  of  the  column  of  nil  A  2?, 
we  shall  have,  p'  +  nw  =  p,    hence, 

«=  %-l£ (150.) 

dg  p' 

Denoting  ihe  modulus  of  the  common  system  of  loga- 
rithms by  31,  and  designating  common  logarithms  by  the 
symbol  log,  we  shall  have, 

Mz  =  §-  log  ^ ,    or    z  =  J?—  log  ^  • 
dg    Gy  Mdg    &  p' 

Now,  the  pressures  jo  and//  are  measured  by  the  heights  of 
the  columns  of  mercury  which  they  will  support;  denoting 
these  heights  by  JET  and  II,  we  have, 

p  _  II 

p '  ~  II' ' 


MECHANICS    OF    GASES    AND    VAPORS.  301 

whence,  by  substitution, 

z  =  iklos^r  •  *  '  (157,) 

We  have  supposed  the  temperature,  both  of  the  air  and 
mercury,  to  be  32°.  In  order  to  make  the  preceding  for- 
mula general,  let  T  represent  the  temperature  of  the  mer- 
cury at  B,  T\  its  temperature  at  aT  and  denote  the  cor- 
responding heights  of  the  barometric  column  by  h  and  hf ; 
also,  let  t  denote  the  temperature  of  the  air  at  B,  and  t'  its 
temperature  at  a. 

t) 

The  quantity        is  the  ratio  of  the  density  of  the  air  at  B, 

to  the  corresponding  pressure,  the  temperature  being  32°. 
According  to  Mariotte's  law,  this  ratio  remains  constant, 
whatever  may  be  the  altitude  of  B  aK*\  o  the  level  of  the 
ocean. 

If  we  denote  the  latitude  of  the  ph^o  by  £,  we  have, 
(Art.  124), 

g  -  g'{\  —  0.002695  cos27,l 

It  has  been  shown,  by  experiment,  that,  vl^n  a  column 
of  mercury  is  heated,  it  increases  in  length  at  thp.  iV-e  of 
t9'90ths  of  its  length  at  32°,  for  each  degree  th^t  the  +^)m- 
perature  is  elevated.     Hence, 


*(>  +  ^?)-* 


K 


T'  -32\         „.,  9990  +  T'-S2 


V    T      9990    / 


9990 


Dividing  the  second  equation  by  the  first,  member  by 
member, 

Ji_        II     9990  -f  T—  32 
h'  ~  II''  9990  +  T—  32  * 


302  MECHANICS. 

TT 

Dividing  both  terms  of  the  fractional  coefficient  of  ■=  by 

the  denominator,  and  neglecting  the  quantity   T  —  32,  in 
comparison  with  9990,  we  have, 

*  =  5£ + ^)  -  a* +-™>  <*•-** 

Whence,  by  reduction, 

H_  _  h_ 1 

H'  "  A'*l  +  .0001  (T-  T')  ' 

The  quantity  z  denotes,  not  only  the  height,  but  also  the 
volume  of  the  column  of  air  aB,  at  32°.  When  the  tem- 
perature is  changed  from  32°,  the  pressures  remaining  the 
same,  this  volume  will  vary,  according  to  the  law  of  Gay 
Lussac. 

If  we  suppose  the  temperature  of  the  entire  column  to  be 
a  mean  between  the  temperatures  at  B  and  cz,  which  we 
may  do  without  sensible  error,  the  height  of  the  column 
will  become,  Equation  (153), 

S  [~1  +  .00204  ^y~    -  32  Yl  =  z[l+.00102(*  +  «'-  64)] 

Hence,  to  adapt  Equation  (157)  to  the  conditions  pro- 
posed, we  must  multiply  the  value  of  2  by  the  factor, 

1  +  .00102(0  +  t'-  64). 

Substituting  in  Equation  (157),  for  —  and  </,  the  values 

shown  above,  and  multiplying  the  resulting  value  of  z,  bv 
the  factor    1  +  .00102(0  +  t'  —  64),    we  have, 


z  — 


p       1  +  .00102(0  4-  t'—  64) 


In, 


Md  1  -  0.002695cos2/         &  A'[l+  .0001  (T-  T')] 

(158.) 


MECHANICS    OF   GASES    AND    VArORS.  303 

P 

The  factor   -  — -  is  constant,  and  may  be  determined  as 

follows:  select  two  points,  one  of  which  is  considerably 
higher  than  the  other,  and  determine,  by  trigonometrical 
measurement,  their  difference  of  level.  At  the  lower  point, 
take  the  reading  of  the  barometer,  of  its  attached  ther- 
mometer, and  of  a  detached  thermometer  exposed  to  the 
air.  Make  similar  observations  at  the  upper  station.  These 
observations,  together  with  the  latitude  of  the  place,  will 
give  all  the  quantities  entering  Equation  (158),  except  the 
factor  in  question.  Hence,  this  factor  may  be  deduced.  It 
is  found  to  be  60345.51  ft.  Hence,  we  have,  finally,  the 
barometric  formula, 

z  —  60345.51  ft.   x 

J   +  .00102  (W'  — 64)     ^  h 

l-0.002695cos2J         °g  A'[l  +.0001(7'-  T')]     * 159*' 


To  use  this  formula  for  determining  the  difference  of  level 
between  two  stations,  observe,  simultaneously,  if  possible, 
the  heights  of  the  barometer  and  of  the  attached  and  de- 
tached thermometers,  at  the  two  stations.  Substitute  these 
results  for  the  corresponding  quantities  in  the  formula ;  also 
substitute  for  I  the  latitude  of  the  place,  and  the  resulting 
value  of  z,  will  be  the  difference  of  level  required. 

If  the  observations  cannot  be  made  simultaneously  at  the 
two  stations,  make  a  set  of  observations  at  the  lower  station  ; 
after  a  certain  interval,  make  a  set  at  the  upper  station ; 
then,  after  an  equal  interval,  make  another  set  at  the  lower 
station.  Take  a  mean  of  the  results  of  observation  at  the 
lower  station,  as  a  single  set,  and  proceed  as  before. 

For  the  more  convenient  application  of  the  formula  for 
the  difference  of  level  between  two  points,  tables  have  been 
computed,  by  means  of  which  the  arithmetical  operations 
are  much  facilitated. 


304 


MECHANICS. 


Work  due  to  the  Expansion  of  a  Gas  or  Vapor. 

201.     Let  the  gas  or  vapor  be  confined  in  a  cylindei 

closed  at  its  lower  end,  and  having 
a  piston  working  air-tight.  When 
the  gas  occupies  a  portion  of  the 
cylinder  whose  height  is  A,  denote  the 
pressure  on  each  square  inch  of  the 
piston  hyp;  when  the  gas  expands, 
so  that  the  altitude  of  the  column  be- 
comes x,  denote  the  pressure  on  a 
square  inch  by  y. 

Since  the  volumes  of  the  gas,  under 
these  suppositions,  are  proportional  to  their  altitudes,  we 
shall  have,  from  Maeiotte's  laws, 


B 


Fig.  172. 


whence 


p  :  y  :  :  x  :  A; 
xy  =  ph 


If  we  suppose  p  and  A  to  be  constant,  and  x  and  y  to 
vary,  the  above  equation  will  be  that  of  an  equilateral 
hyperbola  referred  to  its  asymptotes. 

Draw  AC  perpendicular  to  AM,  and  on  these  lines,  as 
asymptotes,  construct  the  curve  NLII,  from  the  equation, 
xy  =  ph.  .Make  AG  —  A,  and  draw  Gil  parallel  to  AC\ 
it  will  represent  the  pressure  p.  Make  AM  =  x,  and  draw 
J/1V  parallel  to  A  G ;  it  will  represent  the  pressure?/.  In 
like  manner,  the  pressure  at  any  elevation  of  the  piston  may 
be  constructed. 

Let  KL  be  drawn  infinitely  near  to  GIT,  and  parallel 
with  it.  The  elementary  area  GKLII  will  not  differ 
sensibly  from  a  rectangle  whose  base  is  jo,  and  altitude  is 
( i J\.  Hence,  its  area  may  be  taken  as  the  measure  of  the 
work  whilst  the  piston  is  rising  through  the  infinitely  small 
space  (r K.  In  like  manner,  the  area  of  any  infinitely  small 
element,  bounded  by  lines  parallel  to  .1CY,  may  be  taken  to 
represent  the  work  whilst  the  piston  is  rising  through  the 


MECHANICS    OF    GASES    AND    VArORS.  305 

height  of  the  element.  If  we  take  the  sum  of  all  the 
elements  between  the  ordinates  GH  and  JJ/iV,  this  sum,  or 
the  area  G3fJ¥If,  will  represent  the  total  quantity  of  work 
of  the  force  of  expansion  whilst  the  piston  is  rising  from  G 
to  M.  But  the  area  included  between  an  equilateral  hyper- 
bola and  one  of  its  asymptotes,  and  limited  by  lines  parallel 
to  the  other  asymptote,  is  equal  to  the  product  of  the  co- 
ordinates of  any  point,  multiplied  by  the  Naperian 
logarithm  of  the  quotient  obtained  by  dividing  one  of  the 
limiting  ordinates  by  the  other ;  or,  in  this  particular  case, 

it  is  equal  to  ph  x  l(—\     Hence,    if  we   designate    the 

quantity  of  work  performed  by  the  expansive  force  whilst 
the  piston  is  moving  over  &M,  by  ^,  we  shall  have, 


q  =  ph  X  l{^ 

This  is  the  quantity  of  work  exerted  upon  each  square  inch 
of  the  piston ;  if  we  denote  the  area  of  the  piston,  by  A, 
and  the  total  quantity  of  work,  by  (>,  we  shall  have, 

Q-Aphxl(P)  =  Aphxl(^\    .     (160.) 

If  we  denote  by  c  the  number  of  cubic  feet  of  gas,  when  the 

pressure  is  p,  and  suppose  it  to  expand  till  the  pressure  is  y, 

we  shall  have,  Ah  =  c ;    or,  if  A  be   expressed  in  square 

Ah 

feet,  we  shall  have,  c  = Hence,  by  substitution, 

'144  '     J 

Finally,  if  we  suppose  the  pressure  at  the  high<  st  point  to 
he  p',  we  shall  have, 

0.  =  a«qpX/f(Jp), 


306  MECHANICS. 

an  equation  which  gives  the  quantity  of  work  ot  c  cubic 
feet  of  gas,  whilst  expanding  from  a  pressure  p,  to  a  pres- 
sure })'. 

Efflux  of  a  Gas  or  Vapor. 

202.  Suppose  the  gas  to  escape  from  a  small  orifice,  and 
denote  its  velocity  by  v.  Denote  the  weight  of  a  cubic 
foot  of  the  gas,  by  w,  and  the  number  of  cubic  feet  dis- 
charged in  one  second,  by  c,  then  will  the  mass  escaping  in 

cw 
one  second,  be  equal  to  — ,    and  its  living   force  will  be 

cw 
equal  to  — v2.      But,   from  Art.    148,  the   living   force   is 

double  the  accumulated  quantity  of  work.  If,  therefore,  we 
denote  the  accumulated  work  by  §,  we  shall  have, 

r\  cw  2 
Q  =  —v. 
*        2g 

But  the  accumulated  work  is  due  to  the  expansion  of  the 
gas,  and  if  we  denote  the  pressure  within  the  orifice,  by  py 
and  without,  by^',  we  shall  have,  from  Art.  201, 


q  =  lucp  x  i(^y 


Equating  the  second  members,  we  have, 


™v'  =  lUcpxl(£) 


Whence, 


•-»>/¥*<£) 


Substituting   for   g,   its   value,    32i  ft.,    we   have,    aftei 
reduction, 


96 


v^¥)  •  •  •  <i6i> 


MECHANICS   OF   GASES    AND   VArURS.  307 

When  the  difference  between  p  and  p'  is  small,  the  pre- 
ceding formula  can  be  simplified. 

Since  —  =  1  +  ^  ~      ,  we  have,  from  the  logarithmic 
p  p 

series, 

When  p  —p'  is  very  small,  the  second,  and  all  succeeding 
terms  of  the  development,  may  be  neglected,  in  comparison 
with  the  first  term.     Hence, 


© 


p-p 


P 

Substituting,  in  the  formula  above  deduced,  we  have, 


V  to  p 


or,  since  —  is,  under  the  supposition  just  made,  equal  to  1, 
we  have,  finally, 


v  =  96 


v7^ ("*> 


Coefficient  of  Efflux. 

203.  When  air  issues  from  an  orifice,  the  section  of  the 
current  undergoes  a  change  of  form,  analagous  to  the  con- 
traction of  the  vein  in  liquids,  and  for  similar  reasons.  If 
we  denote  the  coefficient  of  efflux,  by  ft,  the  area  of  the 
orifice,  by  A,  and  the  quantity  of  air  delivered  in  n  seconds, 
by  Q,  we  shall  have,  from  Equation  (161), 


Q  =  w^vf^O  • 


MECHANICS. 

According  to  Koch,  the  value  of  k  is  equal  to  .58,  when 
the  orifice  is  in  a  thin  plate  ;  equal  to  .74,  when  the  air 
issues  through  a  tube  6  times  as  long  as  it  is  wide ;  and 
equal  to  .85,  when  it  issues  through  a  conical  nozzle  5  times 
as  long  as  the  diameter  of  the  oritice,  and  whose  sides  have 
a  convergence  of  6°  to  the  axis. 

The  preceding  principles  are  applicable  to  the  distribution 
of  gas,  to  the  construction  of  blowers,  and,  in  general,  to  a 
great  variety  of  pneumatic  machines. 

Steam. 

204.  If  water  be  exposed  to  the  atmosphere,  at  ordinary 
temperatures,  a  portion  is  converted  into  vapor,  which  mixes 
with  the  atmosphere,  constituting  one  of  the  permanent 
elements  of  the  aerial  ocean.  The  tension  of  watery  vapor 
thus  formed,  is  very  slight,  and  the  atmosphere  soon  ceases 
to  absorb  any  more.  If  the  temperature  of  the  water  be 
raised,  an  additional  amount  of  vapor  is  evolved,  and  of 
greater  tension.  When  the  temperature  is  raised  to  that 
point  at  which  the  tension  of  the  vapor  is  equal  to  that  of 
the  atmosphere,  ebullition  commences,  and  the  vaporization 
goes  on  with  great  rapidity.  If  heat  be  added  beyond  the 
point  of  ebullition,  neither  the  water  nor  the  vapor  will 
increase  in  temperature  till  all  of  the  water  is  converted  into 
steam.  When  the  barometer  stands  at  30  inches,  the  boil- 
ing point  of  pure  water  is  212°  Fab.  We  shall  suppose,  in 
what  follows,  that  the  barometer  stands  at  30  inches.  After 
the  temperature  of  the  water  is  raised  to  212°,  the  addi- 
tional heat  that  is  added  becomes  latent  in  the  vapor 
evolved. 

If  heat  be  applied  uniformly,  it  is  found  by  experiment 
thai  it  takes  5i  times  as  much  to  convert  all  of  the  water 
into  Steam  as  it  requires  to  raise  it  from  :",2°  to  212°.  Hence, 
the  entire  amount  of  heat  which  becomes  latent  is 
5^  X  (212°  —  32°)  =  990°.  That  the  heat  applied  becomes 
latent,  may  be  shown  experimentally  as  follows  : 

Let  a  cubic  inch  of  water  be  converted  into  steam  at 


MECHANICS    OF    GASES    AND    VAPORS.  309 

212°,  and  kept  in  a  close  vessel.  Now,  if  5^  cubic  inches 
of  water  at  32°  be  injected  into  the  vessel,  the  steam  will  all 
be  converted  into  water,  and  the  6^  cubic  inches  of  water 
will  be  found  to  have  a  temperature  of  212".  The  heat 
that  was  latent  becomes  sensible  again. 

When  water  is  converted  into  steam  under  any  other 
pressure  than  that  of  the  atmosphere,  or  15  pounds  to  the 
square  inch,  it  is  found  that,  although  the  boiling  point  will 
be  changed,  the  entire  amount  of  heat  required  for  convert- 
ing the  water  into  steam  will  remain  unchanged. 

If  the  evaporation  takes  place  under  such  a  pressure,  that 
the  boiling  point  is  but  150°,  the  amount  of  heat  which 
becomes  latent  is  1052°,  so  that  the  latent  heat  of  the 
steam,  plus  its  sensible  heat,  is  1202°.  If  the  pressure  under 
which  vaporization  takes  place  is  such  as  to  raise  the  boiling 
point  to  500°,  the  amount  of  heat  which  becomes  latent  is 
702°,  the  sum  702°  +  500°  being  equal  to  1202°,  as  before. 
Hence,  we  conclude  that  the  same  amount  of  fuel  is 
required  to  convert  a  given  amount  of  water  into  steam,  no 
matter  what  may  be  the  pressure  under  which  the  evapora- 
tion takes  place. 

When  water  is  converted  into  steam  under  a  pressure  of 
one  atmosphere,  each  cubic  inch  is  expanded  into  about 
1700  cubic  inches  of  steam,  of  the  temperature  of  212°  ;  or, 
since  a  cubic  foot  contains  1728  cubic  inches,  we  may  say, 
in  round  numbers,  that  a  cubic  inch  of  xoater  is  converted 
into  a  cubic  foot  of  steam. 

If  water  is  converted  into  steam  under  a  greater  or  less 
pressure  than  one  atmosphere,  the  density  will  be  increased 
or  diminished,  and,  consequently,  the  volume  will  be  dimin- 
ished or  increased.  The  temperature  being  also  increased 
or  diminished,  the  increase  of  density  or  decrease  of  volume 
will  not  be  exactly  proportional  to  the  increase  of  pressure  ; 
but,  for  purposes  of  approximation,  we  may  consider  the 
densities  as  directly,  and  the  volumes  as  inversely  propor- 
tional to  the  pressures  under  which  the  steam  is  generated. 
Under  this  hypothesis,  if  a  cubic  inch  of  water  be  evapo- 


310  MECHANICS. 

rated  under  a  pressure  of  a  half  atmosphere,  it  will  afford 

two  cubic  feet  of  steam;    if  generated  under  a  pressure  of 

two  atmospheres,  it  will  only  afford  a  half  cubic  foot  of  steam. 

Work  of  Steam. 

205.  When  water  is  converted  into  steam,  a  certain 
amount  of  work  is  generated,  and,  from  what  has  been  shown, 
this  amount  of  work  is  very  nearly  the  same,  whatever  may 
be  the  temperature  at  which  the  water  is  evaporated. 

Suppose  a  cylinder,  whose  cross-section  is  one  square 
inch,  to  contain  a  cubic  inch  of  water,  above  which  is  an  air- 
tight piston,  that  may  be  loaded  with  weights  at  pleasure. 
In  the  first  place,  if  the  piston  is  pressed  down  by  a  weight 
of  15  pounds,  and  the  inch  of  water  converted  into  steam, 
tha  weight  will  be  raised  to  the  height  of  1728  inches,  or 
144  feet.  Hence,  the  quantity  of  work  is  144  x  15,  or, 
2160  units.  Again,  if  the  piston  be  loaded  with  a  weight 
of  30  pounds,  the  conversion  of  water  into  steam  will  give 
but  864  cubic  inches,  and  the  weight  will  be  raised  through 
72  feet.  In  this  case,  the  quantity  of  work  will  be  72  x  30, 
or  2160  units,  as  before.  We  conclude,  therefore,  that  the 
quantity  of  work  is  the  same,  or  nearly  so,  whatever  may  be 
the  pressure  under  which  the  steam  is  generated.  We  also 
conclude,  that  the  quantity  of  work  is  nearly  proportional  to 
the  fuel  consumed. 

Besides  the  quantity  of  work  developed  by  simply  con- 
verting an  amount  of  water  into  steam,  a  further  quantity 
of  work  is  developed  by  allowing  the  steam  to  expand  after 
entering  the  cylinder.  This  principle  is  made  use  of  in 
steam  engines  working  expansively. 

To  find  the  quantity  of  work  developed  by  steam  acting  ex- 
pansively. Let  AJ3  represent  a  cylinder,  closed  at 
A,  and  having  an  air-tight  piston  D.  Suppose  the 
steam  to  enter  at  the  bottom  of  the  cylinder,  and  to 
push  the  piston  upward  to  (7,  and  then  suppose 
the  opening  at  which  the  steam  enters,  to  be 
closed.     If  the  piston  is  not  too  heavily  loaded, 


the  steam  will  continue  to  expand,  and  the  piston        Fig  m 


MECHANICS    OF    GASES    AND    VAPORS.  311 

will  be  raised  to  some  position,  B.  The  expansive  force 
of  the  steam  will  obey  Mariotte's  law,  and  the  quantity  of 
work  due  to  expansion  will  be  given  by  Equation  (  1G0). 

Denote  the  area  of  the  piston  in  square  inches,  by  A  ;  the 
pressure  of  the  steam  on  each  square  inch,  up  to  the  moment 
when  the  communication  is  cut  oft*,  by  p  ;  the  distance  A  C, 
through  which  the  piston  moves  before  the  steam  is  cut  oft*, 
by  A  ;  and  the  distance  AD,  by  nh. 

If  we  denote  the  pressure  on  each  square  inch,  when  the 

piston  arrives  at  J5,  by  p\  we  shall  have,  by  Mariotte's 

law, 

P 
p  :  pf  : :  nh  :  A,       .  • .    p'  —  —  , 

an  expression  which  gives  the  limiting  value  of  the  load  of 
the  piston. 

The  quantity  of  work  due  to  expansion  being  denoted  by 
2,  we  shall  have,  from  Equation  (160), 

q  =  Aph  x  I  (-7- )  —  Aphl  (n)* 


If  we  denote  the  quantity  of  work  of  the  steam,  whilst 
the  piston  is  rising  to  (7,  by  q",  we  shall  have, 

q"  =  Aph. 

Denoting  the  total  quantity  of  work  during  the  entire  stroke 
of  the  piston,  by  Q,  we  shall  have, 

Q  =  Aph[l  -f  l(n)]     .     .     .     (163.) 

Experimental  Formulas. 

206.  Numerous  experiments  have  been  made  for  the 
purpose  of  determining  the  relation  existing  between  the 
elasticity  and  temperature  of  steam  in  contact  with  the 
water  by  which  it  is  produced,  and  many  formulas,  based 


312  MECHANICS. 

upon  these  experiments,  have  been  given,  two  of  which  arc 
subjoined : 

The  formula  of  Duloxg  and  Arago  is, 

p  =  (1  +  .007153*)3, 

in  which  p  represents  the  tension  in  atmospheres,  and  t  the 
excess  of  the  temperature  above  100°  Centigrade. 
Tredgold's  formula  is, 

t  =  0.85  y^  —  75, 

in  which  t  is  the  temperature,  in  degrees  of  the  Centigrade 
thermometer,  and  p  the  pressure,  expressed  in  centimeters 
of  the  mercurial  column. 


HYDRAULIC    AND    PXIXMATIC    MACHINES. 


313 


CHAPTEE  IX. 


HYDRAULIC    AND    PNEUMATIC    MACHINE8. 


Definitions. 

207.  Hydraulic  machines  are  those  used  in  raising  and 
distributing  water,  such  as  pumps,  siphons,  hydraulic  rams, 
&c.  The  name  is  also  applied  to  those  machines  in  which 
water  power  is  the  motor,  or  in  which  water  is  employed  to 
transmit  pressures,  such  as  water-wheels,  hydraulic  presses,  &c. 

Pneumatic  machines  are  those  employed  to  rarefy  and 
condense  air,  or  to  impart  motion  to  the  air,  such  as  air- 
pumps,  ventilating-blowers,  <fcc.  The  name  is  also  applied 
to  those  machines  in  which  currents  of  air  furnish  the  motive 
power,  such  as  windmills,  &c. 

Water  Pumps. 

208.  A  water  pump  is  a  machine  for  raising  water  from 
a  lower  to  a  higher  level,  generally  by  the  aid  of  atmospheric 
pressure.  Three  separate  principles  are  employed  in  the 
working  of  pumps:  the  sucking,  the  lifting,  and  the 
forcing  principle.  Pumps  are  frequently  named  according 
as  one  or  more  of  these  principles  are  employed. 

Sucking  and  Lifting  Pump. 

209.  This  pump  consists  of  a 
cylindrical  barrel  A,  at  the  lower 
extremity  of  which  is  attached  a 
Fucking-pipe  B,  leading  to  a  reser- 
voir. An  air-tight  piston  C  is  work- 
ed up  and  down  in  the  barrel  by 
means  of  a  lever  E,  attached  to  a 
piston-rod  D.  P  represents  a  valve 
opening   upwards,   which,   when    the 

J4 


Fig.  1T4. 


31Jr  MKCIIANICS. 

pump  is  at  rest,  closes  by  its  own  weight.  This  valve 
is  called,  from  its  position,  the  piston-valve.  A  second 
valve  6r,  also  opening  upwards,  is  placed  at  the  junction  of 
the  pipe  with  the  barrel.  This  is  called  the  sleeping-valve. 
The  space  X,M,  through  which  the  piston  can  be  moved  up 
and  down  by  the  lever,  is  called  the  play  of  the  piston. 

To  explain  the  action  of  the  pump,  suppose  the  piston  to 
be  at  the  lowest  limit  of  the  play,  and  everything  in  a  state 
of  equilibrium.  If  the  extremity  of  the  lever  E  be 
depressed,  and  the  piston  consequently  be  raised,  the  air  in 
the  lower  part  of  the  barrel  will  be  rarefied,  and  that  in  the 
pipe  B  will,  by  virtue  of  its  greater  tension,  open  the  valve, 
and  a  portion  of  it  will  escape  into  the  barrel.  The  air  in 
the  pipe,  thus  rarefied,  will  exert  a  less  pressure  upon  the 
water  in  the  reservoir  than  that  of  the  external  air,  and, 
consequently,  the  water  will  rise  in  the  pipe,  until  the  tension 
of  the  internal  air,  plus  the  weight  of  the  column  of  water 
raised,  is  equal  to  the  tension  of  the  external  air;  the  valve 
G  will  then  close  by  its  own  weight. 

If  the  piston  be  again  depressed  to  the  lowest  limit,  by 
means  of  the  lever  E,  the  air  in  the  lower  part  of  the  barrel 
will  be  compressed,  its  tension  will  become  greater  than  that 
of  the  external  air,  the  valve  F  will  be  forced  open,  and  a 
portion  of  the  air  will  escape.  If  the  piston  be  raised  once 
more,  the  water  will,  for  the  same  reason  as  before,  rise  still 
higher  in  the  pipe,  and  after  a  few  double  strokes  of  the 
piston,  the  air  will  be  completely  exhausted  from  beneath 
the  piston,  the  water  will  pass  through  the  piston  valve,  and 
finally  escape  at  the  spout  P. 

The  water  is  raised  to  the  piston  by  the  pressure  of  the 
air  on  the  surface  of  the  water  in  the  reservoir ;  hence,  the 
piston  should  not  be  placed  at  a  greater  distance  above  the 
level  of  the  water  in  the  reservoir,  than  the  height  to  which 
the  pressure  of  the  air  will  sustain  a  column  of  water.  In 
fact,  it  should  be  placed  a  little  lower  than  this  limit.  The 
specific  gravity  of  mercury  being  about  13.5,  the  height  of 
a  column   of  water  which  will  exactly  counterbalance  the 


HYDRAULIC    AND    PNEUMATIC    MACHINES.  315 

pressure  of  the  atmosphere,  will  be  found  by  multiplying  the 
height  of  the  barometric  column  by  13  J. 

At  the  level  of  the  sea  the  average  height  of  the  baro- 
metric  column  is  2\  feet;  hence,  the  theoretical  height  to 
which  water  can  be  raised  by  the  principle  of  suction  alone, 
is  a  little  less  than  34  feet. 

The  water  having  passed  through  the  piston  valve,  it  may 
be  raised  to  any  height  by  the  lifting  principle,  the  only 
limitation  being  the  strength  of  the  pump  and  want  of 
power. 

There  are  certain  relations  which  must  exist  between  the 
play  of  the  piston  and  its  height  above  the  water  in  the 
reservoir,  in  order  that  the  water  may  be  raised  to  the 
piston  ;  for,  if  the  play  is  too  small,  it  will  happen  after  a  few 
strokes  of  the  piston,  that  the  air  between  the  piston  and 
the  surface  of  the  water  will  not  be  sufficiently  compressed 
to  open  the  piston  valve;  when  this  state  of  affairs  takes 
place,  the  water  will  cease  to  rise. 

To  investigate  the  relation  that  must  exist  between  the 
play  and  the  height  of  the  piston  above  the  water. 

Denote  the  play  of  the  piston,  by^>,  the  distance  from  the 
upper  surface  of  the  water  in  the  reservoir  to  the  highest 
position  of  the  piston,  by  «,  and  the  height  at  which  the 
water  ceases  to  rise  in  the  pump,  by  x.  The  distance  from 
the  surface  of  the  water  in  the  pump  to  the  highest  position 
of  the  piston  will  then  be  equal  to  a  —  se,  and  the  distance 
to  the  lowest  position  of  the  piston,  will  be  a  —  p  —  x. 
Denote  the  height  at  which  the  atmospheric  pressure  will 
sustain  a  column  of  water  in  vacuum,  by  A,  and  the  weight  of 
a  column  of  water,  whose  base  is  the  cross-section  of  the 
pump,  and  whose  altitude  is  1,  by  w ;  then  will  wh  denote 
the  pressure  of  the  atmosphere  exerted  upwards  through  the 
water  in  the  reservoir  and  pump. 

Xow,  when  the  piston  is  at  its  lowest  position,  in  order 
that  it  may  not  thrust  open  the  piston  valve  and  escape,  the 
pressure  of  the  confined  air  must  be  exactly  equal  to  that 
of  the  external  atmosphere;  that  is,  equal  to  ich.     When  the 


316  MECHANICS. 

piston  is  at  its  highest  position,  the  confined  air  will  be  rare- 
fied, the  volume  occupied  being  proportional  to  its  height. 
Denoting  the  pressure  of  the  rarefied  air  by  toh\  we  shall 
have  from  Mariotte's  law, 

wh  :  wh'  ::  a  —  x  :  a  —  p  —  x. 
•      wh'  =  wh 


a  —  x 


If  the  water  does  not  rise  when  the  piston  is  at  its  highest 
position,  the  pressure  of  the  rarefied  air,  plus  the  weight  of 
the  column  already  raised,  will  be  equal  to  the  pressure  of 
the  external  atmosphere;  or 

a  —  p  —  x 

wh +  wx  =  wh. 

a  —  x 

Solving  this  equation  with  respect  to  #,  we  have, 


If  we  have, 


_  a  ±  yet?  —  4ph 


Aph  >  a2 ;      or,     p  >  -^  , 


the  value  of  x  will  be  imaginary,  and  there  will  be  no  point 
at  which  the  water  will  cease  to  rise.  Hence,  the,  above 
inequality  expresses  the  relation  that  must  exist,  in  order 
that  the  pump  may  be  eifective.  This  condition  expressed 
in  words,  gives  the  following  rule  : 

The  pump  will  be  effective,  when  the  play  of  the  piston  is 
greater  tl(<m  the  square  of  the  distance  from  the  surface  of 
the  water  in  the  reservoir,  to  the  highest  positio?i  of  the 
piston,  divided  by  four  times  the  height  at  which  the  j)res- 
sure  of  the  atmosphere  will  support  a  column  of  water  in 
a  vacuum. 

Let  it  be  required  to  find  the  least  allowable  play  of  the 
piston,  when  the  highest  position  of  the  piston  is  16  feet 


HYDRAULIC    AND    PNEUMATIC    MACHINES.  317 

above  the  water  in  the  reservoir,  and  when  the  barometer 
stands  at  28  inches. 
In  this  case, 

a  =  16  ft.,      and     h  =  28  in.  x  13-J-  =  378  in.  =  3l£  ft. 

Hence, 

J>>fffft.;      or,    p>2^it. 

To  find  the  quantity  of  work  required  to  make  a  double 
stroke  of  the  piston,  after  the  water  reaches  the  level  of  the 
spout. 

In  depressing  the  piston,  no  force  is  required,  except  that 
necessary  to  overcome  the  inertia  of  the  parts  and  the  fric- 
tion. Neglecting  these  for  the  present,  the  quantity  of 
work  in  the  downward  stroke,  may  be  regarded  as  0.  In 
raising  the  piston,  its  upper  surface  will  be  pressed  down- 
wards, by  the  pressure  of  the  atmosphere  w7i,  plus  the  weight 
of  the  column  of  water  from  the  piston  to  the  spout ;  and  it 
will  be  pressed  upwards,  by  the  pressure  of  the  atmosphere, 
transmitted  through  the  pump,  minus  the  weight  of  a 
column  of  water,  whose  cross-section  is  equal  to  that  of  the 
barrel,  and  whose  altitude  is  the  distance  from  the  piston  to 
the  surface  of  the  water  in  the  reservoir.  If  we  subtract 
the  latter  pressure  from  the  former,  the  difference  will  be 
the  resultant  downward  pressure.  This  difference  will  be 
equal  to  the  weight  of  a  column  of  water,  whose  base  is  the 
cross-section  of  the  barrel,  and  whose  height  is  the  distance 
of  the  spout  above  the  reservoir.  Denoting  the  height  by 
If,  the  pressure  will  be  equal  to  wH.  The  path  through 
which  the  pressure  is  exerted  during  the  ascent  of  the 
piston,  is  equal  to  the  play  of  the  piston,  or  p.  Denoting  the 
quantity  of  work  required,  by  Q,  we  shall  have, 

Q  =  wpll. 

But  wp  is  the  weight  of  a  volume  of  water,  whose  base  is 
the  cross-section  of  the  barrel,  and  whose  altitude  is  the 
play  of  the  piston.     Hence,  the  value  of  Q  is  equal  to  the 


31S 


MECHANICS. 


quantity  of  work  necessary  to  raise  this  volume  of  watei 
from  the  level  of  the  water  in  the  reservoir  to  the  spout. 
This  volume  is  evidently  equal  to  the  volume  actually 
delivered  at  each  double  stroke  of  the  piston.  Hence,  the 
quantity  of  work  expended  in  pumping  with  the  sucking 
and  lifting  pump,  all  hurtful  resistances  being  neglected,  is 
equal  to  the  quantity  of  work  necessary  to  lift  the  amount 
of  water,  actually  delivered,  from  the  level  of  the  water  in 
the  reservoir  to  the  height  of  the  spout.  In  addition  to  this 
work,  a  sufficient  amount  of  power  must  be  exerted,  to 
overcome  the  hurtful  resistances.  The  disadvantage  of  this 
pump,  is  the  irregularity  with  which  the  force  must  act, 
being  0  in  depressing  the  piston,  and  a  maximum  in  raising 
it.  This  is  an  important  objection  when  machinery  is  em- 
ployed in  pumping  ;  but  it  may  be  either  partially  or  entirely 
overcome,  by  using  two  pumps,  so  arranged,  that  the  piston 
of  one  shall  ascend  as  that  of  the  other  descends.  Another 
objection  to  the  use  of  this  kind  of  pump,  is  the  irregularity 
of  flow,  the  inertia  of  the  column  of  water  having  to  be 
overcome  at  each  upward  stroke.  This,  by  creating  shocks, 
consumes  a  portion  of  the  force  applied. 

Sucking  and  Forcing  Pump. 

210.  This  pump  consists  of  a  cylindrical  barrel  A,  with 
its  attached  sucking-pipe  B,  and 
sleeping- valve  #,  as  in  the  pump 
just  discussed.  The  piston  C  is 
solid,  and  is  worked  up  and  down 
in  the  barrel  by  means  of  a  lever 
F,  attached  to  the  piston-rod  D. 
At  the  bottom  of  the  barrel,  a 
branch-pipe  leads  into  an  air-vessel 
K,  tli rough  a  second  sleeping-valve 
J\  which  opens  upwards,  and  closes 
by  its  own  weight.  A  delivery- 
pipe  7/",  enters  the  air-vessel  at  its 
top,  and  terminates  near  its  bottom. 

To   explain    the   action    of   this 


Fig.  178. 


HYDRAULIC    AND    PNEUMATIC    MACHINES.  319 

pump,  suppose  the  piston  C  to  be  depressed  to  its  lowest 
limit.  Now,  if  the  piston  be  raised  to  its  highest  position, 
the  air  in  the  barrel  will  be  rarefied,  its  tension  will  be 
diminished,  the  air  in  the  tube  j5,  will  thrust  open  the  valve, 
and  a  portion  of  it  will  escape  into  the  barrel.  The  pres- 
sure of  the  external  air  will  then  force  a  column  of  water 
up  the  pipe  i?,  until  the  tension  of  the  rarefied  air,  plus  the 
weight  of  the  column  of  water  raised,  is  equal  to  the  tension 
of  the  external  air.  An  equilibrium  being  produced,  the 
valve  G  closes  by  its  own  weight.  If,  now,  the  piston  be 
again  depressed,  the  air  in  the  barrel  will  be  condensed,  its 
tension  will  increase  till  it  becomes  greater  than  that  of  the 
external  air,  when  the  valve  F  will  be  thrust  open,  and  a 
portion  of  it  will  escape  through  the  delivery-pipe  H.  After 
a  few  double  strokes  of  the  piston,  the  water  will  rise 
through  the  valve  G,  and  then,  as  the  piston  descends,  it 
will  be  forced  into  the  air-vessel,  the  air  will  be  condensed 
in  the  upper  part  of  the  vessel,  and,  acting  by  its  elastic 
force,  will  force  a  portion  of  the  water  up  the  delivery-pipe 
and  out  at  the  spout  P.  The  object  of  the  air-vessel  is,  to 
keep  up  a  continued  stream  through  the  pipe  H,  otherwise 
it  would  be  necessary  to  overcome  the  inertia  of  the  entire 
column  of  water  in  the  pipe  at  every  double  stroke.  The 
flow  having  commenced,  at  each  double  stroke,  a  volume  of 
water  will  be  delivered  from  the  spout,  equal  to  that  of  a 
cylinder  whose  base  is  the  area  of  the  piston,  and  whose 
altitude  is  the  play  of  the  piston. 

The  same  relative  conditions  between  the  parts  should 
exist  as  in  the  sucking  and  lifting  pump. 

To  find  the  quantity  of  work  consumed  at  each  double 
stroke,  after  the  flow  has  become  regular,  hurtful  resistances 
being  neglected : 

When  the  piston  is  descending,  it  is  pressed  downwards 
by  the  tension  of  the  air  on  its  upper  surface,  and  upwards 
by  the  tension  of  the  atmosphere,  transmitted  through  the 
delivery-pipe,  plus  the  weight  of  a  column  of  water  whose 
base  is  the  area  of  the  piston,  and  whose  altitude  is  the 


320  MECHANICS. 

distance  of  the  spout  above  the  piston.  This  distance  is 
variable  during  the  stroke,  but  its  mean  vame  is  the  distance 
of  the  middle  of  the  play  below  the  spout.  The  difference 
between  these  pressures  is  exerted  upwards,  and  is  equal  to 
the  weight  of  a  column  of  water  whose  base  is  the  area  of 
the  piston,  and  whose  altitude  is  the  distance  from  the 
middle  of  the  play  to  the  spout.  The  distance  through 
which  the  force  is  exerted,  is  equal  to  the  play  of  the  piston. 
Denoting  the  quantity  of  work  during  the  descending 
stroke,  by  Q'  ;  the  weight  of  a  column  of  water,  having  a 
base  equal  to  the  area  of  the  piston,  and  a  unit  in  altitude, 
by  w;  and  the  height  of  the  spout  above  the  middle  of  the 
the  play,  by  A',  we  shall  have, 

Q'  =  wh'  x  p. 

When  the  piston  is  ascending,  it  is  pressed  downwards 
by  the  tension  of  the  atmosphere  on  its  upper  surface,  and 
upwards  by  the  tension  of  the  atmosphere,  transmitted 
through  the  water  in  the  reservoir  and  pump,  minus  the 
weight  of  a  column  of  water  whose  base  is  the  area  of  the 
piston,  and  whose  altitude  is  the  height  of  the  piston  above 
the  reservoir.  This  height  is  variable,  but  its  mean  value 
is  the  height  of  the  middle  of  the  play  above  the  Mater  in 
the  reservoir.  The  distance  through  which  this  force  is 
exerted,  is  equal  to  the  play  of  the  piston.  Denoting  the 
quantity  of  work  during  the  ascending  stroke,  by  Q'\  and 
the  height  of  the  middle  of  the  play  above  the  reservoir,  by 
A",  we  have, 

Q''  =  wh"  x  p. 

Denoting  the  entire  quantity  of  work  during  a  double  strok 
by  Q,  we  have, 

Q  =   C+  Q"  =  wp{h>  +  h"). 

But  irp  is  the  weight  of  a  volume  of  water,  the  area  of 
whose  base  is  that  of  the  piston,  and  whose  altitude  is  the 


HYDRAULIC    AND    PNEUMATIC    MACHINES. 


321 


play  of  the  piston ;  that  is,  it  is  the  weight  of  the  volume 
delivered  at  the  spout  at  each  double  stroke. 

The  quantity  A'  +  A",  is  the  entire  height  of  the  spout 
above  the  level  of  the  cistern.  Hence,  the  quantity  of  work 
expended,  is  equal  to  that  required  to  raise  the  entire  volume 
delivered,  from  the  level  of  the  water  in  the  reservoir  to  the 
height  of  the  spout.  To  this  must  be  added  the  work 
necessary  to  overcome  the  hurtful  resistances,  such  as  fric- 
tion, &c. 

If  h'  =  A",  we  shall  have,  Q'  =  Q" ;  that  is,  the  quan- 
tity of  work  during  the  ascending  stroke,  will  be  equal  to 
that  during  the  descending  stroke.  Hence,  the  work  of  the 
motor  will  be  more  nearly  uniform,  when  the  middle  of  the 
play  of  the  piston  is  at  equal  distances  from  the  reservoir 
and  spout. 

Fire  Engine. 

211.  The  fire  engine  is  essentially  a  double  sucking  and 
forcing  pump,  the  two  piston  rods  being  so  connected,  that 
when  one  piston  ascends  the  other  descends.  The  sucking 
and  delivery  pipes  are  made  of  some  flexible  material,  gen- 
erally of  leather,  and  are  attached  to  the  machine  by  means 
of  metallic  screw  joints. 

The  figure  exhibits  a  cross-section  of  the  essential  part  of 
a  Fire  Engine. 

A  A'  are  the  two  barrels,  C  C  the  two  pistons,  con- 
nected by  the  rods,  D  D  , 
with  the  lever,  E  E '.  B 
is  the  sucking  pipe,  termi- 
nating in  a  box  from 
which  the  water  may  en- 
ter either  barrel  through 
the  valves,  G  G'.  K  is 
the  air  vessel,  common  to 
both  pumps,  and  com- 
municating with  them  by 
the  valves  F  F '.  II  is 
the  delivery  pipe. 
14* 


Fi«t.  17ft. 


322 


MECHANICS. 


The  instrument  is  mounted  on  wheels  for  convenience  of 
transportation.  The  lever  E  E'  is  worked  by  means  of 
rods  at  right  angles  to  the  lever,  so  arranged  that  several 
men  can  apply  their  strength  in  working  the  pump.  The 
action  of  the  pump  differs  in  no  respect  from  that  of  the 
forcing  pump;  but  when  the  instrument  is  worked  vigor- 
ously,  there  is  more  water  forced  into  the  air  vessel,  the 
tension  of  the  air  is  very  much  augmented,  and  its  elastic 
force,  thus  brought  into  play,  propels  the  water  to  a  consider- 
able distance  from  the  mouth  of  the  delivery  pipe.  It  is 
this  capacity  of  throwing  a  jet  of  water  to  a  great  distance, 
that  gives  to  the  engine  its  value  in  extinguishing  fires. 

A  pump  entirely  similar  to  the  fire  engine  in  its  construc- 
tion, is  often  used  under  the  name  of  the  double  action  forc- 
ing pump  for  raising  water  for  other  purposes. 

The  Rotary  Pump. 

212.  The  rotary  pump  is  a  modification  of  the  sucking 
and  forcing  pump.  Its  construction  will  be  best  understood 
from  the  drawing,  which  represents  a  vertical  section  through 
the  axis  of  the  sucking-pipe,  and  at  right  angles  to  axis  of 
the  rotary  portion  of  the  pump. 

A  represents  an  annular  ring  of  metal,  which  may  be 
made  to  revolve  about  its  axis 
0.  D  D  is  a  second  ring  of 
metal,  concentric  with  the  first, 
and  forming  with  it  an  inter- 
mediate annular  space.  This 
space  communicates  with  the 
sucking-pipe  7f,  and  the  de- 
livery pipe  Z.  Four  radial 
paddles  C\  are  disposed  so  as 
to  slide  backwards  and  for- 
wards through  suitable  open-  Fig.  177. 
ings,   which    are  made    in   the 

ring  A,  and  which  are  moved  around  with  it.      G  is  a  solid 
guide,  firmly  fastened  to  the  end  of  the  cylinder  enclosing 


HYDRAULIC    AND    PNEUMATIC    MACHINES.  323 

the  rotary  apparatus,  and  cut  as  represented  in  the  figure. 
E  E  are  two  springs,  attached  to  the  ring  X>,  and  acting  by 
their  elastic  force,  to  press  the  paddles  firmly  against  the 
guide.  These  springs  are  of  such  dimensions  as  not  to 
impede  the  flow  of  the  water  from  the  pipe  Ji,  and  into  the 
pipe  X. 

When  the  axis  0  is  made  to  revolve,  each  paddle,  as  it 
reaches  and  passes  the  partition  II,  is  pressed  against  the 
guide,  but,  as  it  moves  on,  it  is  forced,  by  the  form  of  the 
guide,  against  the  outer  wall  D.  The  paddle  then  drives 
the  air  in  front  of  it,  around,  in  the  direction  of  the  arrow- 
head, and  finally  expels  it  through  the  pipe  L.  The*  air 
behind  the  paddle  is  rarefied,  and  the  pressure  of  the  exter- 
nal air  forces  a  column  of  water  up  the  pipe.  As  the  paddle 
approaches  the  opening  to  the  pipe  X,  the  paddle  is  pressed 
back  by  the  spring  E,  against  the  guide,  and  an  outlet  into 
the  ascending  pipe  X,  is  thus  provided.  After  a  few  revo- 
lutions, the  air  is  entirely  exhausted  from  the  pipe  K.  The 
water  enters  the  channel  C  C,  and  is  forced  up  the  pipe  X, 
from  which  it  escapes  by  a  spout  at  the  top.  The  quantity 
of  work  expended  in  raising  a  volume  of  water  to  the 
spout,  by  this  pump,  is  equal  to  that  required  to*  lift  it 
through  the  distance  from  the  level  of  the  water  in  the  cis- 
tern to  the  spout.  This  may  be  shown  in  the  same  manner 
as  was  explained  under  the  head  of  the  sucking  and  forcing- 
pump.  To  this  quantity  of  work,  must  be  added  the  work 
necessary  to  overcome  the  hurtful  resistances,  as  fric- 
tion, &c. 

This  pump  is  well  adapted  to  machine  pumping,  the  work 
being  very  nearly  uniform. 

A  machine,  entirely  similar  to  the  rotary  pump,  might  be 
constructed  for  exhausting  foul  air  from  mines ;  or,  by  re- 
versing the  direction  of  rotation,  it  might  be  made  to  force 
a  supply  of  fresh  air  to  the  bottom  of  deep  mines. 

Besides  the  pumps  already  described,  a  great  variety 
of  others  have   been  invented   and   used.      All,  however, 


324 


MECHANICS. 


depend  upon  some  modification  of  the  principles  that  have 
just  been  discussed. 

The  Hydrostatic  Press. 

213.  The  hydrostatic  press  is  a  machine  for  exerting 
great  pressure  through  small  spaces.  It  is  much  used  in 
compressing  seeds  to  obtain  oil,  in  packing  hay  and  bales  of 
goods,  also  in  raising  great  weights.  Its  construction,  though 
requiring  the  use  of  a  sucking-pump,  depends  upon  the  prin- 
ciple of  equal  pressures  (Art.  154). 

It  consists  essentially  of  two  vertical  cylinders,  A  and  B, 
each  provided  with  a  solid  pis- 
ton. The  cylinders  communi- 
cate by  means  of  a  pipe  (7, 
whose  entrance  to  the  larger 
cylinder  is  closed  by  a  sleeping 
valve  E.  The  smaller  cylinder 
communicates  with  the  reser- 
voir of  water  7T,  by  a  sucking- 
pipe  H,  whose  upper  extremity 
is  closed  bythe  sleeping-valve  D. 
The  smaller  piston  7>',  is  worked  up  and  down  by  the  lever 
G.  By  working  the  lever  G,  up  and  down,  the  water  is 
raised  from  the  reservoir  and  forced  into  the  larger  cylinder 
A]  and  when  the  space  below  the  piston  F  is  tilled,  a  force 
of  compression  is  exerted  upwards,  which  is  as  many  times 
greater  than  that  applied  to  the  piston  B,  as  the  area  of 
i^is  greater  than  B  (Art.  L54).  This  force  may  be  util- 
ized in  compressing  a  body  L,  placed  between  the  piston 
and  the  frame  of  the  press. 

Denote  the  area  of  the  larger  piston  by  P,  of  the  smaller, 
by  p,  the  pressure  applied  to  7>,  by/,  and  that  exerted  at 
F,  by  F;  we  shall  have, 


Fi<r.  ITS. 


F:f::P:p, 


.'.     F  = 


P 


If  we  denote  the  longer  arm  of  the  lever  G,  by  X,  and 


HYPIiAULIC    AND    PNEUMATIC    MACHINES.  325 

the  shorter  arm,  by  /,  and  represent  the  force  applied  at  the 
extremity  of  the  longer  irm,  by  K,  we  shall  have  from  the 
principle  of  the  lever  (Art.  78), 

K:f::l:L,  ..f=^. 

Substituting  this  value  of/ above,  we  have, 
^      PKL 


~     pi 

To  illustrate,  let  the  area  of  the  larger  piston  be  100 
square  inches,  that  of  the  smaller  piston  1  square  inch  ;  sup- 
pose the  longer  arm  of  the  lever  to  be  30  inches,  and  the 
shorter  arm  to  be  2  inches,  and  a  force  of  100  pounds  to  be 
applied  at  the  end  of  the  longer  arm  of  the  lever ;  to  find 
the  pressure  exerted  upon  F. 

From  the  conditions, 

P  =  100,    K=  100,    L  =  30,   p  —  1,    and    I  =  2. 

Hence, 

_,        100  X  100  X  30  ,™^  „ 

F  = =   150000  lbs. 

We  have  not  taken  into  account  the  hurtful  resistances, 
hence,  the  total  pressure  of  150000  pounds  must  be  some- 
what diminished. 

The  volume  of  water  forced  from  the  smaller  to  the  larger 
piston,  during  a  single  descent  of  the  piston  F\  will  occupy 
in  the  two  cylinders,  spaces  whose  heights  are  inversely  as 
the  areas  of  the  pistons.  Hence,  the  path,  over  which  f  is 
exerted,  is  to  the  path  over  which  F  is  exerted,  as  P  is  to 
p.     Or,  denoting  these  paths  by  s  and  £,  we  have, 

s:  S::  P:p; 

or,  since   P  :  p  : :  F :  f,   we  si  mil  have, 

8  :  S  :  :  F  :  /,         /.    fa  =  FS. 


326  MECHANICS. 

That  is,  the  quantities  of  icork  of  the  power  and  resistance 
are  equal,  a  principle  which  holds  good  in  all  machines. 

EXAMPLES. 

1.  The  cross-section  of  a  sucking  and  forcing  pump  is  0 
square  feet,  the  play  of  the  piston  3  feet,  and  the  height  of 
the  spout,  above  the  level  of  the  reservoir,  50  feet.  What 
must  be  the  effective  horse  power  of  an  engine  which  can 
impart  30  double  strokes  per  minute,  hurtful  resistances 
being  neglected  ? 

SOLUTION. 

The  number  of  units  of  work  required  to  be  performed 
each  minute,  is  equal  to 

6  X.3  X  50  X  62J  =  56250. 
Hence, 

n    _    5  6  25J)    _    1    _9_3_  !„;, 

n    —    33000    —    X    1.12'        -^"o. 

2.  In  a  hydrostatic  press,  the  areas  of  the  two  pistons  are, 
respectively,  2  and  400  square  Laches,  and  the  two  arms  of 
the  lever  are,  respectively,  1  and  20  inches.  Required  the 
pressure  on  the  larger  piston  for  each  pound  of  pressure 
applied  to  the  longer  arm  of  the  lever  ?  Ans.  4000  lbs. 

3.  The  areas  of  the  two  pistons  of  a  hydrostatic  press 
are,  respectively,  equal  to  3  and  300  square  inches,  and  the 
shorter  arm  of  the  lever  is  one  inch.  What  must  be  the 
length  of  the  longer  arm,  that  a  force  of  1  lb.  may  produce 
a  pressure  of  1000  lbs.  Ans.  10  inches. 

The  Siphon. 

214.     The  siphon  is  a  bent  tube,  used  for  transferring  a 

liquid  from  a  higher  to  a  lower  level,  over  an  in- 
termediate elevation  The  siphon  consists  of  two 
branches,  AB  and  i>C,  of  which  the  outer  one 
is  the  longer.  To  use  the  instrument,  the  tube 
is  filled  with  the  liquid  in  any  manner,  the  end  of 
the  longer  branch  being  stopped  with  the  finger 
or  a  stop-cock,  in  which  case,  the  pressure  of  the 
atmosphere  will  prevent  the  liquid  from  escaping        Fig.  no. 


HYDRAULIC    AND    PNEUMATIC    MACHINFS.  327 

at  the  other  end.  The  instrument  is  then  inverted, 
the  end  C  being  submerged  in  the  liquid,  and  the  stop 
removed  from  A,  The  liquid  will  begin  to  flow  through 
the  tube,  and  the  flow  will  continue  till  the  level  of  the 
liquid  in  the  reservoir  reaches  that  of  the  mouth  of  the 
tube  C. 

To  find  the  velocity  with  which  water  will  issue  from  the 
siphon,  let  us  consider  an  infinitely  small  layer  at  the  orifice 
A  This  layer  will  be  pressed  downwards,  by  the  tension 
of  the  atmosphere  exerted  on  the  surface  of  the  reservoir, 
diminished  by  the  weight  of  the  water  in  the  branch  BD, 
and  increased  by  the  weight  of  the  water  in  the  branch 
BA.  It  will  be  pressed  upwards  by  the  tension  of  the 
atmosjmere  acting  directly  upon  the  layer.  The  difference 
of  these  forces,  is  the  weight  of  the  water  in  the  portion  of 
the  tube  DA,  and  the  velocity  of  the  stratum  will  be  due 
to  that  weight.  Denoting  the  vertical  height  of  DA,  by  h, 
we  shall  have,  for  the  velocity  (Art.  173), 


This  is  the  theoretical  velocity,  but  it  is  never  quite 
realized  in  practice,  on  account  of  resistances,  which  have 
been  neglected  in  the  preceding  investigation. 

The  siphon  may  be  filled  by  applying  the  mouth  to  the 
end  A,  and  exhausting  the  air  by  suction.  The 
tension  of  the  atmosphere,  on  the  upper  surface 
of  the  reservoir,  will  press  the  water  up  the  tube, 
and  fill  it,  after  which  the  flow  will  go  on  as 
before.  Sometimes,  a  sucking-tube  AD,  is  in- 
serted near  the  opening  A,  and  rising  nearly  to 
the  bend  of  the  siphon.  In  this  case,  the  opening 
A,  is  closed,  and  the  air  exhausted  through  the 
sucking-tube  AD,  after  which  the  flow  goes  on  as  before. 

The  Wurtemburg  Siphon. 
215.    In  the  Wurtemburg  siphon,  the  ends  of  the  tube  are 


328 


MECHANICS. 


GZ\ 


m 


<&) 


bent  twice,  at  right-angles,  as  shown  in  the  figure. 
The  advantage  of  this  arrangement  is,  that  the 
tube,  once  filled,  remains  so,  as  long  as  the  plane 
of  its  axis  is  kept  vertical.  The  siphon  may  be 
lifted  out  and  replaced  at  pleasure,  thereby 
stopping  the  flow  at  will. 

It  is  to  be  observed  that  the  siphon  is  only  effectual  when 
the  distance  from  the  highest  point  of  the  tube  to  the  level 
of  the  water  in  the  reservoir  is  less  than  the  height  at  which 
the  atmospheric  pressure  will  sustain  a  column  of  water  in 
a  vacuum.     This  will,  in  general,  be  less  than  3-4  feet. 


Fig.  181. 


The  Intermitting  Siphon. 

216.  The  intermitting  siphon  is  represented  in  the 
figure.  AB  is  a  curved  tube  issuing 
from  the  bottom  of  a  reservoir.  The 
reservoir  is  supplied  with  water  by  a 
tube  E,  having  a  smaller  bore  than 
that  of  the  siphon.  To  explain  its 
action,  suppose  the  reservoir  at  first 
to  be  empty,  and  the  tube  E  to  be 
opened;  as  soon  as  the  reservoir  is 
filled  to  the  level  of  CD,  the  water 
will  begin  to  flow  from  the  opening 

B,  and  the  flow  once  commenced,  will  continue  till  the 
level  of  the  reservoir  is  again  reduced  to  the  level  CD', 
drawn  through  the  opening  A.  The  flow  will  then  cease 
till  the  cistern  is  again  filled  to  CD,  and  so  on  as  before. 


Fie-.  1S2. 


Intermitting  Springs. 

217.  Let  A  represent  a  subterranean  cavity,  communi- 
Cating  with  the  surface  of  the  earth  by 
a  channel  AI><\  bent  like  a  siphon. 
Suppose  the  reservoir  to  lie  fed  by 
percolation  through  the  crevices,  or 
by   a   small    channel   D.     When    the 


HYDRAULIC    AND    PNEUMATIC    MACHINES.  329 

water  in  the  reservoir  rises  to  the  height  of  the  horizontal 
plane  BD,  the  flow  will  commence  at  C,  and,  if  the  chan- 
nel is  sufficiently  large,  the  flow  will  continue  till  the  water 
is  reduced  to  the  level  plane  drawn  through  C.  An  inter- 
mission of  flow  will  occur  till  the  reservoir  is  again  tilled, 
and  so  on,  intermittingly.  This  phenomena  has  been  observed 
at  various  places. 

Siphon  of  Constant  Flow. 

218.  We  have  seen  that  the  velocity  of  efflux  depends 
upon  the  height  of  the  water  in  the  reservoir  above  the 
external  opening  of  the  siphon.  When  the  water  is  drawn 
off  from  the  reservoir,  the  upper  surface  sinks,  this  height 
diminishes,  and,  consequently,  the  velocity  continually 
diminishes. 

If,  however,  the  shorter  branch  <7Z>,  of  the  tube,  be 
inserted  through  a  piece  of  cork  large  enough  to  float  the 
.siphon,  the  instrument  will  sink  as  the  upper  surface  is 
depressed,  the  height  of  DA  will  remain  the  same,  and, 
consequently,  the  flow  will  be  uniform  till  the  bend  of  the 
siphon  comes  in  contact  with  the  upper  edge  of  the  reservoir. 
By  suitably  adjusting  the  siphon  in  the  cork,  the  velocity 
of  efflux  can  be  increased  or  decreased  within  certain  limits. 
In  this  manner,  any  desired  quantity  of  the  fluid  can  be 
drawn  off"  in  a  given  time. 

The  siphon  is  used  in  the  arts,  for  decanting  liquids,  when 
it  is  desirable  not  to  stir  the  sediment  at  the  bottom  of  a 
vessel.  It  is  also  employed  to  draw  a  portion  of  a  liquid 
from  the  interior  of  a  vessel  when  that  liquid  is  overlaid  by 
one  of  less  specific  gravity. 

The  Hydraulic  Ram. 

219.  The  hydraulic  ram  is  a  machine  for  raising  watei 
by  means  of  shocks  caused  by  the  sudden  stoppages  of  a 
stream  of  water. 

The  instrument  consists  of  a  reservoir  .7?,  which  is  sup- 
plied with  water  by  an  inclined  pipe  A  ;  on  the  upper  surface 


330 


MECHANICS. 


Fig.  184 


of  the  reservoir,  is  an  orifice  which  may  be  closed  by 
a  spherical  valve  D\  this  valve, 
when  not  pressed  against  the 
opening,  rests  in  a  metallic 
framework  immediately  below 
the  orifice ;  G  is  an  air-vessel 
communicating  with  the  reser- 
voir by  an  orifice  F,  which  is 
fitted  with  a  spherical  valve  E\ 
this  valve  closes  the  orifice  F, 
except  when  forced  upwards, 
in  which  case  its  motion  is  restrained  by  a  metallic  frame 
work  or  cage ;  //  represents  a  delivery-pipe  entering  the 
air-vessel  at  its  upper  part,  and  terminating  near  the  bot- 
tom. At  P  is  a  small  valve,  opening  inwards,  to  supply 
the  loss  of  air  in  the  air-vessel,  arising  from  absorption  by 
the  water  in  passing  through  the  air  vessel. 

To  explain  the  action  of  the  instrument,  suppose,  at  first, 
that  it  is  empty,  and  all  the  parts  in  equilibrium.  If  a  cur- 
rent of  water  be  admitted  to  the  reservoir,  through  the  in- 
clined pipe  A,  the  reservoir  will  soon  be  filled,  and  com- 
mence rushing  out  at  the  orifice  C.  The  impulse  of  the 
water  will  force  the  spherical  valve  Z>,  upwards,  closing  the 
opening ;  the  velocity  of  the  water  in  the  reservoir  will  be 
suddenly  checked ;  the  reaction  will  force  open  the  valve 
F,  and  a  portion  of  the  water  will  enter  the  air-chamber  G. 
The  force  of  the  shock  having  been  expended,  the  spherical 
valves  will  both  fall  by  their  own  weight ;  a  second  shock 
will  take  place,  as  before ;  an  additional  quantity  of  water 
will  be  forced  into  the  air-vessel,  and  so  on,  indefinite!}'. 
As  the  water  is  forced  up  into  the  air-vessel,  the  air  becomes 
compressed;  and  acting  by  its  elastic  force,  it  urges  a  stream 
of  water  up  the  pipe  //  The  shocks  occur  in  rapid  succes- 
sion, and,  at  each  shock,  a  quantity  of  water  is  forced  into 
the  air-chamber,  and  thus  a  constant  stream  is  kept  up. 
To  explain  the  use  of  the  valvt  ]\  it  maybe  remarked  that 
water  absorbs  more  air  under  a  great  pressure,  than  under 


HYDRAULIC    AND    PNEUMATIC    MACHINES.  331 

a  smaller  one.  Henee,  as  it  passes  through  the  air-chamber, 
a  portion  of  the  air  contained  is  taken  up  by  the  water  and 
carried  out  through  the  pipe  H.  But  each  time  that  the 
valve  D  falls,  there  is  a  tendency  to  produce  a  vacuum 
in  the  upper  part  of  the  reservoir,  in  consequence  of  the 
rush  of  the  fluid  to  escape  through  the  opening.  The  pres- 
sure of  the  external  air  then  forces  the  valve  P  open,  a 
small  portion  of  air  enters,  and  is  afterwards  forced  up  with 
the  water  into  the  vessel  (r,  to  keep  up  the  supply. 

The  hydraulic  ram  is  only  used  where  it  is  required  to 
raise  small  quantities  of  water,  such  as  for  the  supply  of  a 
house,  or  garden.  Only  a  small  fraction  of  the  amount  of 
fluid  which  enters  the  supply-pipe  actually  passes  out 
through  the  delivery-pipe;  but,  if  the  head  of  water  is 
pretty  large,  the  column  may  be  raised  to  a  great  height. 
Water  is  often  raised,  in  this  manner,  to  the  highest  points 
of  lofty  buildings. 

Sometimes,  an  additional  air-vessel  is  introduced  over  the 
valve  E,  for  the  purpose  of  deadening  the  shock  of  the 
valve  in  its  play  up  and  down. 

Archimedes'  Screw. 

220.  This  machine  is  intended  for  raising  water  through 
small  heights,  and  consists,  in  its  simplest  form,  of  a  tube 
wound  spirally  around  a  cylinder.  This  cylinder  is  mounted 
so  that  its  axis  is  oblique  to  the  horizon,  the  lower  end  dip- 
ping into  the  reservoir.  When  the  cylinder  is  turned  on  its 
axis,  by  a  crank  attached  to  its  upper  extremity,  the  lower 
end  of  the  tube  describes  a  circumference  of  a  circle,  whose 
plane  is  perpendicular  to  the  axis.  When  the  mouth  of  the 
tube  comes  to  the  level  of  the  axis  and  begins  to  ascend, 
there  will  be  a  certain  quantity  of  water  in  the  tube,  which  will 
flow  so  as  to  occupy  the  lowest  part  of  the  spire;  and,  if  the 
cylinder  is  properly  inclined  to  the  horizon,  this  flow  will  be 
towards  the  upper  end  of  the  tube.  At  each  revolution,  an 
additional  quantity  of  water  will  enter  the  tube,  and  that 
already  in  the  tube  will  be  forced,  or  raised,  higher  and 


332 


MECHANICS. 


Fig.  1S5. 


higher,  till,  at  last,  it  will  flow  from  the  orifice  at  the  upper 
end  of  the  spiral  tube. 

The  Chain  Pump. 

221.  The  chain  pump  is  an  instrument  for  raising  wate» 
through  small  elevations.     It  consists 

of  an  endless  chain  passing  over  two 
wheels,  A  and  J3,  having  their  axes 
horizontal,  the  one  being  below  the 
surface  of  the  water,  and  the  other 
above  the  spout  of  the  pump.  At- 
tached to  this  chain,  and  at  right 
angles  to  it,  are  a  system  of  circular 
disks,  just  fitting  the  tube  CD.  If 
the  cylinder  A  be  turned  in  the  di- 
rection of  the  arrow-head,  the  buckets 
or  disks  will   rise  through   the  tube 

CD,  carrying  the  water  in  the  tube  before  them,  until  it 
reaches  the  spout  C,  and  escapes.  The  buckets  thus  emptied 
return  through  the  air  to  the  reservoir,  and  so  on  perpetually. 
One  great  objection  to  this  machine  is,  the  difficulty  of 
making  the  buckets  fit  the  tube  of  the  pump.  Hence*  there 
is  a  constant  leakage,  requiring  a  great  additional  expend- 
iture of  force. 

Sometimes,  instead  of  having  the  body  of  the  pump  ver- 
tical, it  is  inclined ;  in  which  case  it  does  not  differ  much 
in  principle  from  the  wheel  with  fiat  buckets,  that  has  been 
used  for  raising  water. 

The  Air  Pump. 

222.  The  air  pump  is  a  machine  for  rarefying  the  air  ir. 
a  closed  space. 

It  consists  of  a  cylindrical 
barrel  A,  in  which  a  piston 
B,  fitting  air-tight,  is  work- 
ed up  and  down  by  a  \e\  ;•:■ 
C\  attached  to  a  piston-rod 
D.  The  barrel  communi- 
cates with  an  air-tight  ves- 


Fig.  186. 


HYDRAULIC    AND    PNEUMATIC    MACHINES.  333 

sel  E,  called  a  receiver,  by  means  of  a  narrow  pipe.  The 
receiver,  which  is  usually  of  glass,  is  ground  so  as  to  fit  air- 
tight upon  a  smooth  bed-plate  KK.  The  joint  between  the 
receiver  and  plate  may  be  rendered  more  perfectly  air-tight 
by  rubbing  it  with  a  little  oil.  A  stop-cock  //,  of  a  peculiar 
construction,  permits  communication  to  be  made  at  pleasure 
between  the  barrel  and  receiver,  or  between  the  barrel  and 
the  external  air.  When  the  stop-cock  is  turned  in  a  partic- 
ular direction,  the  barrel  and  receiver  are  made  to  commu- 
nicate ;  but  on  turning  it  through  90  degrees,  the  communi- 
cation with  the  receiver  is  cut  off,  and  a  communication  is 
opened  between  the  barrel  and  the  external  air.  Instead  of 
the  stop-cock,  valves  are  often  used,  which  are  either  opened 
and  closed  by  the  elastic  force  of  the  air,  or  by  the  force 
that  works  the  pump.  The  communicating  pipe  should  be 
exceedingly  small,  and  the  piston  B  should,  when  at  its  low- 
est point,  fit  accurately  to  the  bottom  of  the  barrel. 

To  explain  the  action  of  the  air  pump,  suppose  the  pLston 
to  be  depressed  to  its  lowest  position.  The  stop-cock  H,  is 
turned  so  as  to  open  a  communication  between  the  barrel 
and  receiver,  and  the  piston  is  raised  to  its  highest  point  by 
a  force  applied  to  the  lever  C.  The  air  which  before  occu- 
pied the  receiver  and  pipe,  will  expand  so  as  to  fill  the  bar- 
rel, receiver,  and  pipe.  The  stop-cock  is  then  turned  so  as  to 
cut  oft'  communication  between  the  barrel  and  receiver,  and 
open  the  barrel  to  the  external  air,  and  the  piston  again  de- 
pressed to  its  lowest  position.  The  rarefied  air  in  the  barrel 
is  expelled  into  the  external  air  by  the  depression  of  the 
piston.  The  air  in  the  receiver  is  now  more  rarefied  than  at 
the  beginning,  and  by  a  continued  repetition  of  the  process 
just  described,  any  degree  of  rarefaction  may  be  attained. 

To  measure  the  degree  of  rarefaction  of  the  air  in  the 
receiver,  a  siphon-gauge  may  be  used,  or  a  glass  tube,  30 
inches  long,  may  be  made  to  communicate  at  its  upper 
extremity  with  the  receiver,  whilst  its  lower  extremity  dips 
into  a  cistern  of  mercury.  As  the  air  is  rarefied  in  the 
receiver,  the  pressure  on  the  mercury  in  the  tube  becomes 


334  MECHANICS. 

less  than  that  on  the  surface  of  the  mercury  in  the  cistern, 
and  the  mercury  rises  in  the  tube.  The  tension  of  the  air 
in  the  receiver  will  be  given  by  the  difference  between  the 
height  of  the  barometric  column  and  that  of  the  mercury 
in  the  tube. 

To  investigate  a  formula  for  computing  the  tension  of  the 
air  in  the  receiver,  after  any  number  of  double  strokes,  let 
us  denote  the  capacity  of  the  receiver  in  cubic  feet,  by  r, 
that  of  the  connecting-pipe,  by  p,  and  the  space  between 
the  bottom  of  the  barrel  and  the  highest  position  of  the 
piston,  by  b.  Denote  the  original  tension  of  the  air,  by  t ; 
its  tension  after  the  first  upward  stroke  of  the  piston,  by  t '; 
after     the     second,    third,    ...»'*,    upward     strokes,    by 

*,  r,  . . .  f. 

The  air  which  originally  occupied  the  receiver  and  pipe, 
fills  the  receiver,  pipe,  and  barrel,  after  the  first  upward 
stroke  ;  according  to  Mariotte's  law,  its  tension  in  the  two 
cases  varies  inversely  as  the  volumes  occupied ;  hence, 


t  •  t'  :  :  p  +  r  +  b  :  p  +  r,  .%  t'  =  t     p  +  f     • 

In  like  manner,  wre  shall   have,  after  the  second    upward 
stroke, 

f  :  t"  :  :  p  +  r  +  b  :  p  +  r,  .-.  t"  =  t' 


p  -f  b  +  r 

Substituting  for  t'  its  value,  deduced  from  the  preceding 
equation,  we  have, 

t"  - 1(  p+r  V 

In  like  manner,  we  find, 

pArt     \s. 


"  =  *(- 

\p 


b  +  r 


HYDRAULIC    AND    PNEUMATIC    MACHINES.  335 

and,  in  general, 

*> = «(.  ;t:  )'• 

If  the  pipe  is  exceedingly  small,  its  capacity  may  be 
neglected  in  comparison  with  that  of  the  receiver,  and  we 
shall  then  have, 


<  =  <jh) 


b  +  r. 

Let  it  be  required,  for  example,  to  determine  the  tension 
of  the  air  after  5  upward  strokes,  when  the  capacity  of  the 
barrel  is  one-third  that  of  the  receiver. 

T 

In  this  case,  ■= =  f,  and  n  —  5,  whence, 

fv    _    f    243    . 

Hence,  the  tension  is  less  than  a  fourth  part  of  that  the 
external  air. 

Instead  of  the  receiver,  the  pipe  may  be  connected  by  a 
screw-joint  with  any  closed  vessel,  as  a  hollow  globe  or  glass 
flask.  In  this  case,  by  reversing  the  direction  of  the  stop- 
cock, in  the  up  and  down  motion  of  the  piston,  the  in- 
strument may  be  used  as  a  condenser.  When  so  used,  the 
tension,  after  n  downward  strokes  of  the  piston,  is  given  by 
the  formula, 


tn' 


m 


Taking  the  same  case  as  that  before  considered,  with  the 
ei  ception  that  the  instrument  is  used  as  a  condenser  instead 
oi  a  rarefier,  we  have,  after  5  downward  strokes, 

fv    /   1  02  4  . 

That  is,  the  tension  is  more  than  four  times  that  of  the 
external  air. 


336  MECHANICS. 

When  the  pump  is  used  for  condensing  air,  it  is  called  a 
condenser. 

Artificial  Fountains. 

223.  An  artificial  fountain  is  an  instrument  by  means  of 
which  a  liquid  is  forced  upwards  in  the  form  of  a  jet,  by 
the  tension  of  condensed  air.  The  simplest  form  of  an  arti- 
ficial fountain  is  called  Hero's  ball. 

Hero's  Ball. 

224.  This  instrument  consists  of  a  hollow  globe  A,  into 
the  top  of  which  is  inserted  a  vertical  tube  J5, 

reaching  nearly  to  the  bottom  of  the  globe. 
This  tube  is  provided  with  a  stop-cock  C\  by 
means  of  which  it  may  be  closed,  or  opened  to 
the  external  air,  at  pleasure.  A  second  tube 
J),  enters  the  globe  near  the  top,  which  is  also 
provided  with  a  stop-cock  E. 

To  use  the  instrument,  close  the  stop-cock  (7,  Fig.  187. 

and  fill  the  lower  portion  of  the  globe  with 
water  through  the  tube  D ;  then  attach  the  tube  D  to  a 
condenser,  and  pump  air  into  the  upper  part  of  the  globe, 
and  confine  it  there  by  closing  the  stop-cock  E.  If,  now,  the 
stop-cock  C  be  opened,  the  pressure  of  the  confined  air  on 
the  surface  of  the  water  in  the  globe,  will  force  a  jet  up 
through  the  tube  B.  This  jet  will  rise  to  a  greater  or  less 
height,  according  to  the  greater  or  less  quantity  of  air  that 
was  forced  into  the  globe.  The  water  will  continue  to  flow 
through  the  tube  as  lon<r  as  the  tension  of  the  confined  air 
is  greater  than  that  of  the  external  atmosphere,  or  else  till 
the  level  of  the  water  in  the  globe  reaches  the  lower  end 
of  the  tube. 

Instead  of  using  the  condenser,  air  may  be  introduced  by 
blowing  with  the  mouth  through  the  tube  Z>,  and  then  con- 
fined as  before,  by  turning  the  stop-cock  E. 

The  principle  of  IIeko's  ball  is  the  same  as  that  of  the  air- 
chamber  in  the  forcing  pump  and  fire-engine,  already  ex- 
plained. 


HYDRAULIC    AND    PNEUMATIC    MACHINE8. 


337 


i 


3d 


Fig.  188 


Hero's  Fountain. 

225.  Hero's  fountain  is  constructed  on  the  same  prin- 
ciple as  Hero's  ball,  except  that  the  compression  of  the  air 
is  effected  by  the  weight  of  a  column  of  water,  instead  of  by 
aid  of  a  condenser 

A  represents  a  cistern,  similar  to  Hero's  ball,  with  a  tube 
J5,  extending  nearly  to  the  bottom  of  the  cis- 
tern. C  is  a  second  cistern  placed  at  some 
distance  below  A.  This  cistern  is  connected 
with  a  basin  D,  by  a  bent  tube  E,  and  also 
with  the  upper  part  of  the  cistern  A,  by  a 
tube  F.  When  the  fountain  is  to  be  used, 
the  cistern  A  is  nearly  filled  with  water, 
the  cistern  C  being  empty.  A  quantity  of 
water  is  then  poured  into  the  basin  _Z),  which, 
acting  by  its  weight,  sinks  into  the  cistern  C, 
compressing  the  air  in  the  upper  portion  of  it 
into  a  smaller  space,  thus  increasing  its  tension. 
This  increase  of  tension  acting  on  the  surface 
of  the  water  in  A,  forces  a  jet  through  the  tube  J5,  which 
rises  to  a  greater  or  less  height  according  to  the  greater  or 
less  increase  of  the  atmospheric  tension.  The  flow  will  con- 
tinue till  the  level  of  the  water  in  A,  reaches  the  bottom  of 
the  tube  B.  The  measure  of  the  compressing  force  on  a 
unit  of  surface  of  the  water  in  G7,  is  the  weight  of  a  column 
of  water,  whose  base  is  a  square  unit,  and  whose  altitude  is 
the  difference  of  level  between  the  water  in  D  and  C. 

If  Hero's  ball  be  partially  filled  with  water  and  placed 
under  the  receiver  of  an  air  pump,  the  water  will  be  ob- 
served to  rise  in  the  tube,  forming  a  fountain,  as  the  air  in 
the  receiver  is  exhausted.  The  principle  is  the  same  as 
before,  an  excess  of  pressure  on  the  water  within  the  globe 
over  that  without.  In  both  cases,  the  flow  is  resisted  by  the 
tension  of  the  air  without,  and  is  urged  on  by  the  tension 

within. 

Wine-Taster  and  Dropping-Bottle. 

226.  The  wine-taster  is  used  to  bring  up  a  small  por- 


338  MECHANICS. 

tion  of  wine  or  other  liquid,  from  a  cask.  It 
consists  of  a  tube,  open  at  the  top,  and  terminat- 
ing below  in  a  very  narrow  tube,  also  open.  When 
it  is  to  be  used,  it  is  inserted  to  any  depth  in  the 
liquid,  which  will  rise  in  the  tube  to  the  level  of 
the  upper  surface  of  that  liquid.  The  finger  is 
then  placed  so  as  to  close  the  upper  orifice  of 
the  tube,  and  the  instrument  is  raised  out  of  the  !g' 
cask.  A  portion  of  the  fluid  escapes  from  the  lower  orifice, 
until  the  pressure  of  the  rarefied  air  in  the  tube,  plus  the 
weight  of  a  column  of  liquid,  whose  cross-section  is  that  of 
the  tube,  and  whose  altitude  is  that  of  the  column  of  fluid 
retained,  is  just  equal  to  the  pressure  of  the  external  air. 
If  the  tube  be  placed  over  a  tumbler,  and  the  finger  re- 
moved from  the  upper  orifice,  the  fluid  brought  up  will 
escape  into  the  tumbler. 

If  the  lower  orifice  is  very  small,  a  few  drops  may  be 
allowed  to  escape,  by  taking  off  the  finger  and  immediately 
replacing  it.     The  instrument  then  constitutes  the  dropping 

tube. 

The  Atmospheric  Inkstand. 

227.     The  atmospheric  inkstand  consists  of  a  cylinder 
A,  which   communicates  by  a   tube  with  a 
second  cylinder  B.     A  piston   C,  is  moved  JP 

up  and  down  in  A,  by  means  of  a  screw  D. 
Suppose  the  spaces  A  and  B,  to  be  filled 
with  ink.  If  the  piston  G  is  raised,  the 
pressure  of  the  external  air  forces  the  ink  to 
follow  it,  and  the  part  B  is  emptied.     If  the  Fig.  lto 

operation    be    reversed,    and    the    piston    C 
depressed,  the  ink  is  again  forced  into  the  space  B.    This 
operation  may  be  repeated  at  pleasure. 


rc 


APPENDIX. 


The  following  notes  contain  elementary  demonstrations 
of  those  principles,  which  in  the  body  of  the  work  are 
proved  by  means  of  the  Calculus. 

Note  on  Articles  64—70  ;  pp.  72—76. 

These  articles  may  be  omitted  without  at  all  impairing 
the  unity  of  the  subject,  the  preceding  principles  being  suf- 
ficient to  find  the  centre  of  gravity  of  all  bodies,  approxima- 
tively. 

Note  on  Articles  112—114;  pp.  143—148. 

The  principal  formulas  in  these  articles  may  be  deduced 
as  follows : 

112.  By  definition,  a  body  moves  uniformly  when  it 
passes  over  equal  spaces  in  equal  times;  now  if  it  passes 
over  a  space  v  in  one  second,  it  will  pass  over  t  times  that 
space  in  t  seconds ;  that  is,  it  will  pass  over  a  space  vt.  If 
we  suppose  it  to  have  passed  over  a  space  s'  before  the  com- 
mencement of  the  time  t,  we  shall  have  for  the  entire  space 
passed  over,  and  which  may  be  denoted  by  s, 

8  =  Vt  +  %* (58.) 

This  equation  corresponds  to  Equation  (58)  of  the  text. 

113.  The  formulas  of  Article  113  may  be  omitted  with- 
out impairing  the  unity  of  the  course.  They  are  only  of 
use  in  Higher  Mechanics,  where  the  employment  of  the  Cal- 
culus is  a  necessity. 


340  MECHANICS. 

114.  Uniformly  varied  motion,  is  that  in  which  the 
velocity  increases  or  diminishes  uniformly.  In  the  former 
case  the  motion  is  accelerated,  in  the  latter  it  is  retarded. 
In  both  cases  the  moving  force  is  constant. 

Denote  the  moving  force  by/*,  the  mass  moved  being  the 
unit  of  mass. 

According  to  Art.  24,  the  measure  of  the  force  is  the  ve- 
locity impressed  in  a  unit  of  time,  that  is,  in  1  second.  Now 
from  the  principal  of  inertia,  Art.  18,  it  follows  that  a  force 
will  produce  the  same  general  effect  upon  a  body,  whethei 
it  finds  the  body  at  rest  or  in  motion.  Hence,  the  velocity 
impressed  in  any  second  of  time  is  constant ;  that  is,  if  the 
velocity  impressed  in  one  second  of  time  is  f  in  t  seconds 
it  will  be  t  times  f  or  ft.  Denoting  the  velocity  by  v, 
we  shall  have, 

v  =ft (69.) 

If  the  body  has  a  velocity  v'  at  the  beginning  of  the  time 
t,  this  velocity  is  called  the  initial  velocity.  Adding  this  to 
the  velocity  imparted  during  the  time  t,  we  have, 

V  =  v'  +  ft (67.) 

With  respect  to  the  space  passed  over,  it  may  be  re- 
marked that  the  velocity  increases  uniformly ;  hence  the 
space  passed  over  in  any  time,  is  the  same  that  it  would 
have  passed  over  in  the  same  time,  had  it  moved  uniformly 
during  that  time  with  its  mean  or  average  velocity.  Now, 
if  a  body  start  from  a  state  of  rest,  its  velocity  at  starting  is 
0,  and  at  the  end  of  the  time  t  it  is  ft,  Equation  (69)  ;  the 
average  or  mean  of  these  is  \ft.  But  the  space  described 
in  the  time  t,  when  the  body  moves  with  the  uniform  ve- 
locity \ft,  is  (Equation  55)  equal  to  \ft  x  t ;  denoting 
the  space  by  *,  we  have, 

s   =   ift* (70.) 

Kin  Equation  (70),  we  make    t  —  \,    we  have, 
s  -  \f\     or,    f=   2s; 


APPENDIX.  341 

that  is,  if  a  body  moves  from  a  state  of  rest,  the  space  de- 
scribed in  the  first  second  of  time,  is  equal  to  half  the 
measure  of  the  accelerating  force ;  or,  the  acceleration  is 
measured  by  twice  the  space  passed  over  in  one  second  of 
time. 

If  we  suppose  that  a  body  starts  from  rest  before  the  be- 
ginning of  the  tinie  t,  so  as  to  pass  over  a  space  sr  before 
the  beginning  of  t,  it  will  during  that  time  have  acquired 
some  velocity,  which  we  may  denote  by  v'.  The  space 
reckoned  from  the  origin  of  spaces  up  to  the  position  of  the 
body  at  the  end  of  the  time  £,  is  made  up  of  three  parts ; 
first,  the  space  s\  called  the  initial  space  ;  second,  a  space 
due  to  the  velocity  v'  during  the  time  t,  which  is  measured 
by  v't;  third,  a  space  due  to  the  action  of  the  incessant  force 
during  the  time  t,  which  will  (Equation  70)  be  equal  to 
ift2.    Adding  these  together,  we  have  finally, 

8   =   *'+  V't+  \f&       .      .      .      (68.) 

If,  in  Equations  (67)  and  (68),  we  suppose  /to  be  essen- 
tially  positive,  the  motion  will  be  accelerated ;  if  we  suppose 
it  to  be  essentially  negative,  the  motion  will  be  retarded, 
and  these  equations  become 

v  =  v'  -ft (VI.) 

s  =  8>+v't-  \f&  .     .     .     .     (72.) 

Note  on  Article  121,  pp.  163—164. 

The  formula  deduced  in  the  first  part  of  this  article  is 
needed  in  the  investigations  of  Acoustics  and  Optics,  and 
can  only  be  found  by  the  Calculus  This  part  of  the  article 
may  be  omitted  without  impairing  the  unity  of  the  course. 

Note  on  Article  123,  pp.  166—168. 

This  article,  up  to  the  end  of  Equation  (95),  may  be  re- 
placed  by  the  following  demonstration: 


U2 


MECHANICS. 


The  simple  pendulum. 

123.  A  pendulum  is  a  heavy  body  suspended  from  a 
horizontal  axis  about  which  it  is  free  to  vibrate. 

In  order  to  investigate  the  circumstances  of  vibration,  let 
us  first  consider  the  hypothetical  case  of  a  single  material 
point,  vibrating  about  an  axis  to  which  it  is  attached  by  a 
rod  destitute  of  weight.  Such  a  pendulum  is  called  a 
simple  pendulum.  The  laws  of  vibration  in  this  case  will 
be  identical  with  those  explained  in  Art.  120,  the  arc  ABC 
being  an  arc  of  a  circle. 

Let  AB  C  be  the  arc  through 
which  the  vibration  takes  place, 
and  denote  its  radius  DA,  by  I. 
The  angle  ABC  is  called  the 
amplitude  of  vibration  ;  half  of 
this  angle,  ABB,  is  called  the 
angle  of  deviation. 

If  the  point  starts  from  rest  at 
A,  it  will,  on  reaching  any  point 
If,  have  a  velocity  v,  due  to  the 
height  EK,  denoted  by  A,  (Art. 
120).     Hence, 


(92.) 


Let  us  suppose  that  the  angle  of  deviation  is  so  small,  that 
the  chords  of  the  arcs  AB  and  HB,  may  be  considered 
equal  to  the  arcs  themselves.  We  shall  have  (Davies'  Le- 
gendre,  Bk.  IV.,  Prop.  XXIIL,  Cor.), 

AB2  =   21  x  EB,     and     1W  =  21  x  KB, 

whence,  by  subtraction, 

AB2  -  IIB2  -   2l(EB  -  KB)  =  21  x  A. 


APPENDIX.  34-3 

Denoting  AB  by  a,    and   HB  by  jc,  and  solving  the 
last  equation,  we  have, 


21 
Substituting  this  value  of  h  in  (92)  it  becomes, 


v  =  yJJ{a?  -  a?)     ....     (a.) 


Now  let  us  develop  the  arc  ABC  into  a  straight  line 
.4'.Z?'  0",  and  suppose  a  material  point  to  start  from  A'  at 
the  same  time  that  the  pendulum  starts  from  A,  and  to 
vibrate  back  and  forth  upon  A'B'  C  with  the  same  veloci- 
ties as  the  pendulum ;  then,  when  the  pendulum  is  at  any 
point  JET,  this  material  point  will  be  at  the  corresponding 
point  H\  and  the  times  of  vibration  of  the  two  will  be 
exactly  the  same. 

To  find  the  time  of  vibration  along  the  line  A'B'C,  de- 
scribe upon  it  a  semi-circle  A'3fC,  and  suppose  a  third 
material  point  to  start  from  A'  at  the  same  time  as  the 
second,  and  to  move  uniformly  around  the  arc  with  a  ve- 
locity equal  to  a  \J  j  •      Then  will  the  time  required  for 

this  particle  to  reach  C  be  equal  to  the  space  divided  by 
the  velocity  (Art.  112).  Denoting  this  time  by  t,  and  re- 
membering that    A'B'  =  a,    we  shall  have, 


g  v  g 


Make  ITB'  =  x,  and  draw  WM  perpendicular  to  A'  C\ 
and  at  M  decompose  the  velocity  of  the  third  particle 
MT  into  two  components  3IN~  and  MQ,  respectively  par- 
allel and  perpendicular  to  A'C. 


344  mecha:sics. 

We  shall  have  for  the  horizontal  component  JIN", 

MN  =  JIT  cos  TMN. 

But,  JUT  =  «\/f,  and  because  JIT  and  JAY  are  re- 
spectively perpendicular  to  B ' JI  and  II' JI,  we  have, 
cos  TJIX  =    cos  B'JIIF  =    ~^L-        But    JB'Jf  =  a, 


and    II'JI  —   J  a?  —  #2 ;    hence,    cos  TJIX  = 

v  a 

Substituting   these   values   in   Equation    (5),  we   have   for 
the  horizontal  velocity, 


Mir  =  y^  -  »?), 


which  is  the  same  value  as  that  obtained  for  v  in  Equa- 
tion (a).  Hence,  we  infer  that  the  velocity  of  the  third 
material  point  in  the  direction  of  A'  C  is  always  equal  to 
that  of  the  second  point,  consequently  the  times  required 
to  pass  from  A'  to  C  must  be  equal ;  that  is,  the  time 
of  vibration  of  the  second  point,  and  consequently  of  the 

pendulum,  must  be  ttyj  -  •  Denoting  this  time  by  f,  we 
have, 

<  =  V? {95-] 

Note  on  Article  131,  pp.  182—186. 

This  article  may  be  omitted  without  impairing  the  unity 
of  the  course.  The  results  may  be  assumed  if  needed. 
They  can  only  be  deduced  by  the  Calculus  by  demon- 
strations too  tedious  for  an  Elementary  Course. 


THE  NATIONAL   SERIES   OF  STANDARD   SCHOOL-BOOKS. 


MATHEMATICS. 


DAYIES'S   COMPLETE   SERIES- 

ARITHMETIC. 

Davies'  Primary  Arithmetic. 

Davies'  Intellectual  Arithmetic. 

Davies'  Elements  of  Written  Arithmetic. 

Davies'  Practical  Arithmetic. 

Davies'  University  Arithmetic. 

TWO-BOOK    SERIES. 

First  Book  in  Arithmetic,   Primary  and  Mental. 
Complete  Arithmetic. 

ALGEBRA. 
Davies'   New  Elementary  Algebra. 
Davies'   University  Algebra. 
Davies'   New  Bourdon's  Algebra. 

GEOMETRY. 
Davies'  Elementary  Geometry  and  Trigonometry. 
Davies'   Legendre's  Geometry. 
Davies'  Analytical  Geometry   and  Calculus. 
Davies'   Descriptive   Geometry. 
Davies'   New  Calculus. 

MENSURATION. 
Davies'  Practical  Mathematics  and  Mensuration. 
Davies'  Elements  of   Surveying. 
Davies'   Shades,   Shadows,   and  Perspective. 

MATHEMATICAL    SCIENCE. 

Davies'   Grammar  of  Arithmetic. 

Davies'   Outlines  of   Mathematical  Science. 

Davies'   Nature  and  Utility  of   Mathematics. 

Davies'    Metric    System. 

Davies  &  Peck's  Dictionary  of  Mathematics. 

17 


THE   NATIONAL    SERIES   OF   STANDARD   SCHOOL-BOOKS. 

DAVIES'S   NATIONAL   COURSE 
OF   MATHEMATICS. 

ITS    RECORD. 

In  claiming  for  this  series  the  first  place  among  American  text-books,  of  whatever 
class,  the  publishers  appeal  to  the  magnificent  record  which  its  volumes  have  earneq 
during  the  thirty-fine  years  of  Dr.  Charles  Davies's  mathematical  labors.  The  unremit- 
ting exertions  of  a  life-time  have  placed  tins  modem  series  on  the  same  proud  eminence 
among  competitors  that  each  of  its  predecessors  had  successively  enjoyed  in  a  course  of 
constantly  improved  editions,  now  rounded  to  their  perfect  fruition,  —  for  it  seems 
almost  that  this  science  is  susceptible  of  no  further  demonstration. 

During  the  period  alluded  to,  many  authors  and  editors  in  this  department  Iipvb 
started  into  public  notice,  and,  by  borrowing  ideas  and  processes  original  with  Dr.  Davies, 
have  enjoyed  a  brief  popularity,  but  are  now  almost  unknown.  Many  of  the  series  of 
to-day,  built  upon  a  similar  basis,  and  described  as  "  modern  books,"  are  destined  to  a 
similar  fate;  while  the  most  far-seeing  eye  will  find  it  difficult  to  fix  the  time,  on  the 
basis  of  any  data  afforded  by  their  past  history,  when  these  books  will  cease  to  increase 
and  prosper,  and  lix  a  still  (inner  hold  on  the  affection  of  every  educated  American. 

One  cause  of  this  unparalleled  popularity  is  found  in  the  fact  that  the  enterprise  of  the 
author  did  not  cease  with  the  original  completion  of  his  books.  Always  a  practical 
teacher,  he  has  incorporated  in  his  text-books  from  time  to  time  the  advantages  or  every 
improvement  in  methods  of  teaching,  and  every  advance  in  science.  During  all  the 
years  in  which  he  has  been  laboring  lie  constantly  submitted  his  own  theories  ami  those 
of  others  to  the  practical  test  of  the  class-room,  approving,  rejecting,  or  modifying 
them  as  the  experience  thus  obtained  might  suggest.  In  this  way  he  has  been  aide 
to  produce  an  almost  perfect  series  of  class-books,  in  which  every  department  of 
mathematics  lias  received  minute  and  exhaustive  attention. 

Upon  the  death  of  Dr.  Davies,  which  took  place  in  1876,  his  work  was  immediately 
taken  up  by  his  former  pupil  and  mathematical  associate  of  many  years,  Prof.  W.  G. 
Peck,  L.L.D.,  of  Columbia  College.  By  him,  with  Prof.  J.  H.  Van  Amringe,  of  Columbia 
College,  the  original  series  is  kept  carefully  revised  and  up  to  the  times. 


Davies's  System  is  the  ACKNOWLEDGES  National  Stan-dard  for  the  United 
States,  for  the  following  reasons:  — 

1st,  It  is  the  basis  nf  instruction  in  the  great  national  schools  at  West  Point  and 
Annapolis. 

2d.     It  has  received  the  quasi  indorsement  of  the  National  Congress. 

3d.     It  is  exclusively  used  in  the  public  schools  of  the  National  Capital 

4th.  The  officials  of  the  Government  use  it  as  authority  in  all  cases  Involving  mathe- 
matical questions. 

5th.  Our  great  soldiers  and  sailors  commanding  the  national  armies  and  navies  were 
e  lucated  in  tins  system.  So  have  been  a  majority  of  eminent  scientists  in  this  country. 
All  these  refer  to  "  Davies"  as  authority. 

6th.  A  larger  number  of  American  citizens  have  received  their  education  from  (his 
than  from  any  other  - 

7th.  The  series  has  a  larger  circulation  throughout  the  whole  coun  fay  than  any  other, 
being  extensively  used  in  every  State  in  the  Union. 

It 


THE   NATIONAL    SERIES    OF   STANDARD    SCHOOL-BOOKS. 
DAVIES     AND      PECK'S     ARITHMETICS. 

OPTIONAL   OR   CONSECUTIVE. 

The  best  thoughts  of  these  two  illustrious  mathematicians  are  combined  in  the 
following  beautiful  works,  which  are  tlie  natural  successors  of  Davies's  Arithmetics, 
sumptuously  printed,  and  bound  in  crimson,  green,  and  gold:  — 

Davies  and  Peck's  Brief  Arithmetic. 

Also  called  the  "  Elementary  Arithmetic."  It  is  the  shortest  presentation  of  the  sub- 
ject, and  is  adequate  for  all  grades  in  common  schools,  being  a  thorough  introduction  to 
practical  life,  except  for  the  specialist. 

At  first  the  authors  play  with  the  little  learner  for  a  few  lessons,  by  object-teaching 
and  kindred  allurements  ;  but  he  soon  begins  to  realize  that  study  is  earnest,  as  he 
becomes  familiar  with  the  simpler  operations,  and  is  delighted  lo  hud  himself  master  of 
important  results. 

The  second  part  reviews  the  Fundamental  Operations  on  a  scale  proportioned  to 
the  enlarged  intelligence  of  the  learner.  It  establishes  the  General  Principles  and 
Properties  of  Numbers,  ami  then  ]  loceeds  to  Fractions.  Currency  and  the  Metric 
System  are  fully  treated  in  connet  won  with  Decimals.  Compound  Numbers  and  Re- 
duction follow,  and  finally  Percentage  with  all  its  varied  appli cations. 

An  Index  of  words  and  principles  concludes  the  book,  for  which  every  scholar  and 
most  teachers  will  be  grateful.  How  much  time  has  been  spent  in  searching  for  a  half- 
forgotten  definition  or  principle  in  a  former  lesson  ! 

Davies  and  Peck's  Complete  Arithmetic. 

This  work  certainly  deserves  its  name  in  the  best  sense.  Though  complete,  it  is  not, 
like  most  others  which  bear  the  same  title,  cumbersome.  These  authors  excel  in  clear, 
lucid  demonstrations,  teaching  the  science  pure  and  simple,  yet  not  ignoring  convenient 
methods  and  practical  applications. 

For  turning  out  a  thorough  business  man  no  other  work  is  so  well  adapted.  He  will 
have  a  clear  comprehension  of  the  science  as  a  whole,  and  a  working  acquaintance 
with  details  which  must  serve  him  well  in  all  emergencies.  Distinguishing  features  of 
the  book  are  the  logical  progression  of  the  subjects  and  the  great  variety  of  practical 
problems,  not  puzzles,  which  are  beneath  the  dignity  of  educational  science.  A  clear- 
minded  critic  has  said  of  Dr.  Peck's  work  that  it  is  free  from  that  juggling  with 
numbers  which  some  authors  falsely  call  "  Analysis."  A  series  of  Tables  for  converting 
ordinary  weights  and  measures  into  the  Metric  System  appear  in  the  later  editions. 


PECK'S    ARITHMETICS. 
Peck's  First  Lessons  in  Numbers. 

This  book  begins  with  pictorial  illustrations,  and  unfolds  gradually  the  science  of 
numbers.  It  noticeably  simplifies  the  subject  by  developing  the  principles  of  addition 
and  subtraction  simultaneously  ;  as  it  does,  also,  those  of  multiplication  and  division. 

Peck's  Manual  of  Arithmetic. 

This  book  is  designed  especially  'or  those  who  seek  sufficient  instruction  to  carry 
them  successfully  through  practical  life,  but  have  not  time  for  extended  study. 

Peck's  Complete  Arithmetic. 

This  completes  the  series  but  is  a  much  briefer  book  than  most  of  the  complete 
arithmetics,  and  is  recommended  not  only  for  what  it  contains,  but  also  for  what  is 
omitted. 

It  may  be  said  of  Dr.  Peck's  books  more  truly  than  of  any  other  series  published,  that 
they  are  clear  and  simple  in  definition  and  rule,  and  that"  superfluous  matter  of  every 
kind  has  been  faithfully  eliminated,  thus  magnifying  the  working  value  of  the  book 
and  saving  unnecessary  expense  of  time  and  labor. 

19 


THE   NATIONAL    SERIES   OF   STANDARD   SCHOOL-BOOKS. 


Algebra.  The  student's  progress  in  Algebra  depends  very  Largely npon  the  proper  treat- 
ment of  the  four  Fundamental  Operation*.  The  terms  Addition,  Subtraction,  Multiplication, 
and  Jjirision  in  Algebra  have  a  wider  meaning  than  in  Arithmetic,  and  these  operations 
have  been  so  defined  as  to  include  their  arithmetical  meaning  ;  so  that  the  beginner 
is  sinrily  called  upon  to  enlarge  his  views  of  those  fundamental  operations.  Much 
attention  has  been  given  to  the  explanation  of  the  negative  sign,  in  order  to  remove  the 
well-known  difficulties  in  the.  use  and  interpretation  of  that  sign.  Special  attention  is 
here  called  to  "  A  Short  Method  of  Removing  Symbols  of  Aggregation,"  Art  76.  On 
account  of  their  importance,  the  BUOJects  ol  Factoring,  Greatrst  Common  lHrist>r,  and 
Least  Common  Multiple  have  been  treated  al  greater  length  than  is  usual  in  elementary 
works.  In  the  treatment  of  Fractions,  a  method  is  used  which  is  quite  simple,  and, 
at  the  same  time,  more  general  than  that  usually  employed.  In  connection  with  Radical 
s  the  roots  are  expressed  by  fractional  exponents,  for  the  principles  and  rules 
applicable  to  integral  exponents  may  then  be  used  without  modification.  The  Equation 
is  made  the  chief  subject  of  thought  in  this  work.  It  is  defined  near  the  beginning, 
and  used  extensively  in  every  chapter.  In  addition  to  this,  four  chapters  are  devoted 
exclusively  to  the  subject  of  Equations.  All  Proportions  are  equations,  and  in  their 
treatment  as  such  all  the  difficulty  commonly  connected  with  the  subject  of  Proportion 
disappears.  The  chapter  on  Logarithms  will  doubtless  be  acceptable  to  many  teachers 
who  do  not  require  the  student  to  master  Higher  Algebra  before  entering  upon  the 
study  of  Trigonometry. 


HIGHER     MATHEMATICS. 
Peck's  Manual  of  Algebra. 

Bringing  the  methods  of  Bourdon  within  the  range  of  the  Academic  Course. 

Peck's  Manual  of  Geometry. 

By  a  method  purely  practical,  and  unembarrassed  by  the  details  which  rather  confuse 
than  simplify  science. 

Peck's  Practical  Calculus. 

Peck's  Analytical  Geometry. 

Peck's  Elementary  Mechanics. 

Peck's  Mechanics,  with  Calculus. 

The  briefest  treatises  on  these  subjects  now  published.  Adopted  by  the  great  Univer- 
sities :  rale,  Harvard,  Columbia,  Princeton,  Cornell,  &e 

Macnie's  Algebraical  Equations. 

Serving  as  a  e plemenl  to  the  more  advanced  treatises  on  Algebra,  giving  special 

attention  to  the  analysis  and  solution  of  equations  with  numerical  coefficients. 

Church's  Elements  of  Calculus. 

Church's  Analytical  Geometry. 

Church's  Descriptive  Geometry.     "With  plates.     £  vois. 

These  volumes  constitute  the  "West  Point  Course"  in  their  several  departments. 
Pr< it.  Church  was  long  the  eminent  professor  of  mathematics  at  West  Point  Military 
Academy,  and  his  works  are  standard  in  all  the  leading  colleges. 

Courtenay's  Elements  of  Calculus. 

A  standard  work  of  the  very  highest  grade,  presenting  the  most  elaborate  attainable 
butv(  y  ol 

Hackley's  Trigonometry. 

With  applications  to  Navigation  and  Surveying,  Nautical  and  Practical  Geometry, 
and  Geodesy. 

21 


THE  NATIONAL    SERIES   OF  STANDARD   SCHOOL-BOOKS. 


'7\-;?3ssi 

1 

HvS 

i|/    1 

L^i^Jj^J^ 

GENERAL    HISTORY. 

Monteith's  Youth's  History  of  the  United  States. 

A  History  of  the  United  States  for  beginners.  It  is  arranged  upon  the  catechetical  plan, 
with  illustrative  maps  and  engravings,  review  questions,  dates  in  parentheses  (that  their 
study  may  be  optional  with  the  younger  class  of  learners),  and  interesting  biographical 
sketches  of  all  persons  who  have  been  prominently  identified  with  the  history  of  our 
country. 

Willard's  United   States.      School  and  University  Editions. 

The  plan  of  this  standard  work  is  chronologically  exhibited  in  front  of  the  titlepage. 
The  maps  and  sketches  are  found  useful  assistants  "to  the  memory  ;  and  dates,  usually 
so  difficult  to  remember,  are  so  systematically  arranged  as  in  a  great  degree  to  obviate 
the  difficulty.  Candor,  impartiality,  and  accuracy  are  the  distinguishing  features  of 
the  narrative  portion. 

Willard's  Universal  History.     New  Edition. 

The  most  valuable  features  of  the  li  United  States  "  are  reproduced  in  this.  The 
peculiarities  of  the  work  are  its  great  conciseness  and  the  prominence  given  to  the 
chronological  order  of  events.  The  margin  marks  each  successive  era  with  great  dis- 
tinctness, so  that  the  pupil  retains  not  only  the  event  but  its  time,  and  thus  fixes  the 
order  of  history  firmly  and  usefully  in  his  mind.  Mrs.  Willard's  books  are  constantly 
revised,  and  at  all  times  written  up  to  embrace  important  historical  events  of  recent 
date.     Professor  Arthur  Gilman  has  edited  the  last  twenty-five  years  to  1882. 

Lancaster's  English   History. 

By  the  Master  of  the  Stoughton  Grammar  School,  Boston.  The  most  practical  of  the 
"brief  books."  Though  short,  it  is  not  a  bare  and  uninteresting  outline,  but  contains 
enough  of  explanation  and  detail  to  make  intelligible  the  cause  mid  effect  of  events. 
Their  relations  to  the  history  and  development  of  the  American  people  is  made  specially 
prominent. 

Willis's  Historical  Reader. 

Being  Collier's  Great  Events  of  History  adapted  to  American  schools.  This  rare 
epitome  of  general  history,  remarkable  for  its  charming  style  and  judicious  selection  of 
events  on  which  the  destinies  of  nations  have  turned,  has  been  skilfully  manipulated 
by  Professor  Willis,  with  as  few  changes  as  would  bring  the  United  States  into  its  ]  in  iper 
position  in  the  historical  perspective.  As  reader  or  text-book  it  has  few  equals  and  no 
superior. 

Berard's  History  of  England. 

By  an  authoress  well  known   for  the  success  of  her  History  of  the  United  States. 
•  I  life  of  the  English  people  is  felicitously  interwoven,  as  in  fact,  with  the  civil 
and  military  transactions  of  the  realm. 

Ricord's  History  of  Rome. 

Possesses  the  charm  of  an  romance.     The  fables  with  which  this  history 

abounds  are  introduced  in  such  a  way  as  not  to  deceive  the  inexperienced,  while  adding 

materially  to  the  value  of  the  work  as  t  reliable  index  to  the  character  and  institutions, 
as  well  as  the  historv  of  the  Roman  people. 


THE   NATIONAL    SERIES   OF   STANDARD    SCHOOL-BOOKS. 


vwrte 


m 
m 


\& 


*  M 


■i 


* 


l  i?  if 


;  ]•  f  1 


A  Brief  History  of  An- 
cient Peoples. 

With  an  account  of  their  monuments, 
literature,  and    manners.     340 
12mo.     Profusely  illustrated. 

In  this  work  the  political  history, 
which  occupies  nearly,  if  not  all, 
the  ordinary  school  text,  is  con 
to  the  salient  and  essential  facts,  in 
order  to  give  room  for  a  clear  outline 
of  the  literature,  religion,  architecture, 
character,  habits,  &c,  of  each  nation. 
Surely  it  is  as  important  to  know  some- 
thing aboui  Plato  as  all  about  Caesar, 
and  to  learn  how  the  ancients  wrote 
their  hooks  as  how  they  fought  their 

battles. 

The  chapters  OH  Manners  and  Cus- 
toms ami  the  Scenes  in  Real  Life  repre- 
sent the  people  of  history  as  men  and 
women  subject,  tnthc  same  wants,  hopes 
ami  fears  as  ourselves,  and  so  brine  the  distant  past  near  to  us.  The  Scenes,  which  are 
intended  only  for  reading,  are  the  result  of  a  careful  study  of  the  unequalled  collect  ions  oj 
monuments  in  the  London  and  Berlin  Museums,  of  the  ruins  in  Rome  and  Pompeii,  and 
of  the  latest  authorities  on  the  domestic  life  of  ancient  peoples.  Though  intei  I 
written  in  a  semi-romantic  style,  they  are  accurate  pictures  of  what  might  have  occurred, 
and  some  of  them  are  simple  transcriptions  of  the  details  sculptured  in  Assyrian 
alabaster  or  painted  on  Egyptian  walls. 

36 


THE   NATIONAL    SERIES   OF  STANDARD    SCHOOL-BOOKS. 


HISTORY  —  Continued. 

The  extracts  made  from  the  sacred  books  of  the  East  are  not  specimens  of  their  style 
and  teachings,  but  only  gems  selected  often  from  a  mass  of  matter,  much  of  which  would 
be  absurd,  meaningless,  and  even  revolting.  It  has  not  seemed  best  to  cumber  a  book 
like  this  with  selections  conveying  no  moral  lesson. 

Ihe  numerous  cross-references,  the  abundant  dates  in  parenthesis,  the  pronunciation 
of  the  names  in  the  Index,  the  choice  reading  references  at  the  close  of  each  general 
subject,  and  the  novel  Historical  Recreations  in  the  Appendix,  will  be  of  service  to 
teacher  and  pupil  alike. 

Though  designed  primarily  for  a  text-book,  a  large  class  of  persons  — general  readers, 
who  desire  to  know  something  about  the  progress  of  historic  criticism  and  Aie  recent 
discoveries  made  among  the  resurrected  monuments  of  the  East,  but  have  no  leisure  to 
read  the  ponderous  volumes  of  Brugsch,  Layard,  Grote,  Mommsen,  and  lime  —  will  tiud 
this  volume  just  what  they  need. 


From  Homer  B.    Spkague,  HeaJ  Master 
Girls'  High  School,  If 'est  Newton  St. ,  Bos- 
ton, Muss. 
"  I  beg  to  recommend  in  strong  terms 

the    adoption    of   Barnes's   'History    of 


Ancient  Peoples'  as  a  text-book.  It  is 
about  as  nearly  perfect  as  could  be 
hoped  for.  The  adoption  would  give 
great  relish  to  the  study  of  Ancient 
History." 


HE  Brief  History  of  France. 


By  the  author  of  the  "  Bri  f  United  States." 
with  all  the  attractive  features  of  that  popu- 
u  work  (which  see)  and  new  ones  of  its  own. 

It  is  believed  that  the  History  of  France 
has  never  before  been  presented  in  such 
brief  compass,  and  this  is  effected  without 
sacrificing  one  particle  of  interest.  The  book 
reads  like  a  romance,  and,  while  drawing  tht 
student  by  an  irresistible  fascination  to  his 
task,  impresses  the  great  outlines  indelibly  upon  the  memory. 

27 


THE  NATIONAL    SERIES   OF   STANDARD   SCHOOL-BOOKS. 


DRAWING. 

BARNES'S     POPULAR     DRAWING     SERIES. 

Based  upon  the  experience  of  the  most  successful  teachers  of  drawing  in  the  United 
States. 

The  Primary  Course,  consisting  of  a  manual,  ten  cards,  and  three  primary 
Irawing  I is,  a.  B.  and  U. 

Intermediate  Course.     Four  numbers  and  a  manual. 

Advanced  Course.     Four  numbers  and  a  manual. 

Instrumental  Course.     Pour  numbers  and  a  manual. 

iiie  Intermediate,  Advanced,  and  Instrumental  Courses  are  furnished  either  in  book 
or  <-ar>l  lorm  at  the  same  prici  s.  The  books  contain  tin-usual  blanks,  with  the  unusual 
advantage  of  opening  from  the  pupil,  —  placing  the  copy  directly  in  front  and  above 
the  blank,  thus  occupying  hut  little  desk-room.  The  cards  are  in  the  end  more  econom- 
ical than  the  books,  if  used  in  connection  with  the  patent  blank  folios  that  accompany 
this  series. 

The  cards  are  arranged  to  be  bound  (or  tied)  in  the  folios  and  removed  at  pleasure. 
The  pupil  at  the  end  of  each  number  has  a  complete  book,  containing  only  his  own 
work,  while  the  copies  are  preserved  and  inserted  in  another  fol.o  ready  for  use  in  the 
next  class. 

Patent  Blank  Folios.  No.  1.  Adapted  to  Intermediate  Course.  No.  2.  Adapted 
to  Advanced  and  Instrumental  Courses. 

ADVANTAGES   OF  THIS   SERIES. 

The  Plan  and  Arrangement.  —  The  examples  are  so  arranged  that  teachers  and 
pupils  can  see,  at  a  glance,  how  they  are  to  be  treated  and  where  they  are  to  be  copied. 
In  this  system,  copying  and  designing  do  not  receive  all  the  attention.  The  plan  is 
broader  in  its  aims,  dealing  with  drawing  as  a  branch  of  common-school  instruction, 
asd  giving  it  a  wide  educational  value. 

Correct  Methods.  —  In  this  system  the  pupil  is  led  to  rely  upon  himself,  and  not 
upon  delusive  mechanical  aids,  as  printed  guide-marks,  &c. 

One  of  the  principal  objects  of  any  good  course  in  freehand  drawing  is  to  educate  the 
eye  to  estimate,  location,  form,  and  size.  A  system  which  weakens  the  motive  or  re- 
moves the  necessity  Of  thinking  is  false  in  theory  and  ruinous  in  practice.  The  object 
should  be  to  educate,  not  cram  ;  to  develop  the  intelligence,  not  teach  tricks. 

Artistic  Effect  —The  beautj  of  the  examples  is  not  destroyed  by  crowding  the 
pages  with  useless  and  badly  printed  text.  The  Manuals  contain  all  necessary 
instruction. 

Stages  of  Development. —Many  of  the  examples  are  accompanied  by  diagrams, 
showing  the  different  stag*  -  of  development 

Lithographed  Examples.  —  The  examples  are.  printed  in  imitation  of  pencil 
drawing  (uoi  in  hard,  blach  lines)  that  the  pupil's  work  may  resemble  them. 

One  Term's  Work.  —  Each  book  contains  what  can  be  accomplished  in  an  average 
term,  and  no  more.    Thus  a  pupil  finishes  one  book  before  beginning  another. 

Quality —  not  Quantity.  —.Success  in  drawing  depends  upon  the  amount  of  thought 
exercised  by  the  pupil,  and  not  upon  the  large  number  of  examples  drawn. 

Designing.  —  Elementary  design  is  more  skilfully  taught  in  this  system  than  by 
any  other,  m  addition  to  the  instruction  given  in  the  books,  the  pupil  will  and  printed 
on  the  inside-  of  the  covers  a  variety  of  beautiful  patterns. 

Enlargement  and  Reduction*.  —  The  practice  of  enlarging  and  reducing  from 
cop.es  is  not  commenced  mini  the  pupil  is  well  advanced  in  the  course  and  therefore 
better  able  to  cope  with  this  difficult  feature  in  drawing. 

Natural  Forms.  -This  is  the  only  course  that  gives  at  convenient  intervals  easy 
and  progressive  exercises  in  the  drawing  of  natural  forms. 

Economy.  —  By  the  patent  binding  described  above,  the  copies  need  no!  be  tin-own 
at  de  when  a  book  is  filled  out,  but  arc  preserved  in  perfeel  condition  for  future  use. 
'ihe  blank  books,  only,  will  have  to  De  purchased  after tlu  first  introduction,  th 
ing  of  nmre  than  half  in  the  usual  cost  of  drawing-books. 

Manuals  for  Teachers.  —  The  Manuals  accompanying  this  series  contain  pra<  tical 
g  in  the  class-room,  with  definite  directions  for  draw- 
ing each  of  the  examples  in  the  books,  instructions  lor  designing,  model  and  object 
tlrawing,  drawing  from  natural  forms,  &c. 

28 


THE  NATIONAL    SERIES   OF   STANDARD    SCHOOL-BOOKS. 
DRAWING  —  Continued. 

Chapman's  American  £)rawing-Book. 

The  standard  American  text-book  and  authority  in  all  branches  of  art.  A  compilation 
of  art  principles.  A  manual  for  the  amateur,  and  basis  of  study  for  the  professional 
artist.     Adapted  for  schools  and  private  instruction. 

Contents.  —  "Any  one  who  can  Learn  to  Write  can  Learn  to  Draw." —  Primary  In- 
struction in  Drawing.  —  Rudiments  of  Drawing  the  Human '  Head.  —  Rudiments  in 
Drawing  the  Human  Figure.  —  Rudiments  of  Drawing.  —  The  f  Geometry.  - 

Ferspective.  —  Of  Studying  and  Sketching  from  Nature.  —Of  Painting.  —  Etching  and 
En-raving. —  Of  Modelling. — Of  Composition.  —  Advice  to  the  American  Art-Student. 

The  work  is  of  course  magnificently  illustrated  with  all  the  original  designs. 

Chapman's  Elementary  Drawing-Book. 

A  progressive  course  of  practical  exercises,  or  a  text-book  for  the  training  of  the 
eye  and  hand.  It  contains  the  elements  from  the  larger  work,  and  a  copy  should  be  in 
the  hands  of  every  pupil  ;  while  a  copy  of  the  "  American  Drawing- Book,"  named  above, 
should  be  at  hand  for  reference  by  the  class. 

Clark's  Elements  of  Drawing. 

A  complete  course  in  this  graceful  art,  from  the  first  rudiments  of  outline  to  the 
finished  sketches  of  landscape  and  scenery. 

Allen's  Map-Drawing  and  Scale. 

This  method  introduces  a  new  era  in  map-drawing,  for  the  following  reasons  :  1.  It 
is  a  system.  This  is  its  greatest  merit.  —  2.  It  is  easily  understood  and  taught.  — 
3.  The  eye  is  trained  to  exact  measurement  by  the  use  of  a  scale.  — 4.  By  no  special 
effort  of  the  memory,  distance  and  comparative  size  are  fixed  in  the  mind.  —  5.  It  dis- 
cards useless  construction  of  lines. — 6.  It  can  be  taught  by  any  teacher,  even  though 
there  may  have  been  no  previous  practice  in  map-drawing. — 7.  Any  pupil  old  enough 
to  study  geography  can  learn  by  this  system,  in  a  short  time,  to  draw  accurate  maps. 
—  8.  The  system  is  not  the  result  of  theory,  but  comes  directly  from  the  school-room. 
It  has  been  thoroughly  and  successfully  tested  there,  with  all  grades  of  pupils.  —  9.  It 
is  economical,  as  it  requires  no  mapping  plates.  It  gives  the  pupil  the  ability  of  rapidly 
drawing  accurate  maps. 

FINE     ARTS. 

Hamerton's  Art  Essays  (Atlas  Series)  :  — 

No.  1.    The  Practical  Work  of  Painting. 

With  portrait  of  Rubens.     Svo.     Paper  covers. 

No.  2.    Modern  Schools  of  Art- 
Including  American,  English,  and  Continental  Painting.    Svo.     Paper  covers. 

Huntington's  Manual  of  the  Fine  Arts. 

A  careful  manual  of  instruction  in  the  history  of  art,  up  to  the  present  time. 

Boyd's  Karnes'  Elements  of  Criticism. 

The  best  edition  of  the  best  work  on  art  and  literary  criticism  ever  produced  in 
English. 

Benedict's  Tour  Through  Europe. 

A  valuable  companion  for  anyone  wishing  to  visit  the  galleries  and  sights  of  the 
continent  of  Europe,  as  well  as  a  charming  book  of  travels. 

Dwight's  Mythology. 

A  knowledge  of  mythology  is  necessary  to  an  appreciation  of  ancient  art. 

Walker's  World's  Fair. 

The  industrial  and  artistic  display  at  the  Centennial  Exhibition. 

29 


THE  NATIONAL   SERIES   OF  STANDARD   SCHOOL-BOOKS. 

DR.  STEELE'S  OJME-TERM  SERIES, 
IN  ALL  THE  SCIENCES. 

Steele's  14-Weeks  Course  in  Chemistry. 
Steele's  14-Weeks  Course  in  Astronomy. 
Steele's  14-Weeks  Course  in  Physics. 
Steele's  14-Weeks  Course  in  Geology. 
Steele's  14-Weeks  Course  in  Physiology. 
Steele's  14-Weeks  Course  in  Zoology. 
Steele's  14-Weeks  Course  in  Botany. 

Our  text-books  in  these  studies  are,  as  a  general  thing,  dull  and  uninteresting. 
They  contain  from  400  to  GOO  pages  of  dry  facts  and  unconnected  details.  They  abound 
in  that  which  the  student  cannot  learn,  much  less  remember.  The  pupil  commences 
the  study,  is  confused  by  the  hue  print  and  coarse  print,  and  neither  knowing  exactly 
what  to  learn  nor  what  to  hasten  over,  is  crowded  through  the  single  term  generally 
assigned  to  each  branch,  and  frequently  comes  to  the  close  without  a  definite  and  exact 
idea  of  a  single  scientific  principle. 

Steele's  '•  Fourteen.  Weeks  Courses"  contain  only  that  which  every  well-informed  per- 
son should  know,  while  all  that  which  concerns  only  the  professional  scientist  is  omitted. 
The  language  is  clear,  simple,  and  interesting,  and  the  illustrations  bring  the  subject 
within  the  range  of  home  life  and  daily  experience.  They  give  such  of  the  general 
principles  and  the  prominent  facts  as  a  pupil  can  make  familiar  as  household  words 
within  a  single  term.  The  type  is  large  and  open;  there  is  no  fine  print  to  annoy ; 
the  cuts  are  copies  of  genuine  experiments  or  natural  phenomena,  and  are  of  tine 
execution. 

In  line,  by  a  system  of  condensation  peculiarly  his  own.  the  author  reduces  each 
branch  to  the  limits  of  a  single  term  of  study,  while  sacrificing  nothing  that  is  essential, 
and  nothing  that  is  usually  retained  from  the  study  of  the  larger  manuals  in  common 
use.  Thus  the  student  has  rare  opportunity  to  economize  his  time,  or  rather  to  employ 
that  which  he  has  to  the  best  advantage. 

A  notable  feature  is  the  author's  charming  "style,"  fortified  by  an  enthusiasm  over 
his  subject  in  which  the  student  will  not  fail  to  partake.  Believing  that  Natural 
Science  is  full  of  fascination,  he  has  moulded  it  into  a  form  that  attracts  the  attention 
and  kindles  the  enthusiasm  of  the  pupil. 

The  recent  editions  contain  the  author's  "  Practical  Questions"  on  a  plan  never 
before  attempted  in  scientific  text-books.  These  are  questions  as  to  the  nature  and 
cause  of  common  phenomena,  and  are  not  directly  answered  in  the  text,  the  design 
being  to  test  and  promote  an  intelligent  use  of  the  student's  knowledge  of  the  foregoiDg 
principles. 

Steele's  Key  to  all  His  Works. 

This  work  is  mainly  composed  of  answers  to  the  Practical  Questions,  and  solutions  of  the 
problems,  In  the  author's  celebrated  "  Pourteen-Weeks  Courses  "  in  the  several  sciences, 
with  many  hints  to  teachers,  minor  tables,  fee.    Should  he  on  every  teacher's  desk. 

Prof.  J.  Dorman  Steele  is  an  indefatigable  student,  as  well  as  author,  and  his  books 
have  reached  a  fabulous  circulation.  It  is  sate  to  say  of  his  hooks  that  they  have 
accomplished  more  tangible  and  better  results  in  the  class-room  than  any  other  ever 
offered  to  American  schools,  and  have  been  translated  into  more  languages  for  foreign 
schools.     They  are  even  produced  in  raised  type  for  the  blind. 

32 


\ 


THE  NATIONAL    SERIES   OF  STANDARD   SCHOOL-BOOKS. 


NATURAL    SCIENCE  —  Continued. 


BOTANY. 

Wood's  Object-Lessons  in  Botany. 
Wood's  American  Botanist  and  Florist. 
Wood's  New  Class-Book  of  Botany. 

The  standard  text-hooks  of  the  United  States  in  this  department.  In  style  they  are 
simple,  popular,  and  lively  ;  in  arrangement,  easy  and  natural  ;  in  description,  graphic 
and  scientific.     The  Tables  for  Analysis  are  reduced  to  a  perfect  system.     They  include 

the  flora  of  the  whole  Unit<id  States  east  of  the  Rocky  Mountaius,  and  are  well  adapted 
to  the' regions  west. 

Wood's  Descriptive  Botany. 

A  complete  flora  of  all  plants  growing  east  of  the  Mississippi  River. 

Wood's  Illustrated  Plant  Record. 

A  simple  form  of  blanks  for  recording  observations  in  the  field. 

Wood's  Botanical  Apparatus. 

A  portable  trunk,  containing  drying  pie's,  knir'e,  trowel,  microscope,  and  tweezers, 
and  a  copy  of  Wood's  "  Plant  Record,"  —  the  collector's  complete  outfit. 

Willis's  Flora  of  New  Jersey. 

The  most  useful  book  of  reference  e  <  cr  published  for  collectors  in  all  parts  of  the 
country.  It  contains  also  a  Botanical  Directory,  with  addresses  of  living  American 
botanists. 

Young's  Familiar  Lessons  in  Botany. 

Combining  simplicity  of  diction  with  some  degree  of  technical  and  scientific  knowl- 
edge, lor  intermediate  classes.    Specially  adapted  for  the  Southwest. 


Wood  &  Steele's  Botany. 

See  page  33. 


AGRICULTURE. 

Pendleton's  Scientific  Agriculture. 

A  text-book  for  colleges  and  schools  ;  treats  of  the  following  topics  :  Anatomy  and 
Physiology  of  Plants  ;  Agricultural  Meteorology  ;  Soils  as  related  to  Physics  ;  Chemistry 
of  the  Atmosphere  ;  of  Plants  ;  of  Soils  ;  Fertilizers  and  Natural  Manures  ;  Animal  Nu- 
trition, &c.     By  E.  M.  Pendleton,  M.  D.,   Professor  of  Agriculture  in  the  University  of 

Georgia. 


From  President  A.  D.  White,  Cornell 
University. 
"Dear  Sir:  I  have  examined  your 
'Text-book  of  Agricultural  Science,'  and  it 
seems  to  me  excellent  in  view  of  the  pur- 
pose it  is  intended  to  serve.  Many  of 
your  chapters  interested  me  especially, 
and  all  parts  of  the  work  seem  to  combine 
scientific  instruction  witli  practical  infor- 
mation in  proportions  dictated  by  sound 
common  sense." 


From  President  Robinson,  of  Brown 
University. 
"  It  is  scientific  in  method  as  well  as  in 
matter,  comprehensive  in  plan,  natural 
and  logical  in  order,  compact  and  lucid  in 
irs  statements,  and  must  be  Useful  both  as 
a  text-book  in  agricultural  colleges,  and 
as  a  hand-book  for  intelligent  planters  and 
farmers." 


37 


THE   NATIONAL    SERIES   OF   STANDARD   SCHOOL-BOOKS. 
NATURAL  SCIENCE— Continued. 

PHYSIOLOGY. 

Jarvis's  Elements  of  Physiology. 
Jarvis's  Physiology  and  Laws  of  Health. 

The  only  books  extant  which  approach  this  subject  with  a  proper  view  of  the  true 
object  of  teaching  Physiology  in  schools,  viz.,  that  scholars  may  know  how  to  take  care 
of  their  own  health."  In  hold  contrast  with  the  abstract  Anatomies,  which  children 

learn  as  they  would  Greek  or  Latin  (and  forget  as  soon),  to  discipline  the  mind,  are  these 
text-hooks,  using  the  science  as  a  secondary  consideration,  and  only  so  far  as  is  neces- 
sary lor  the  comprehension  of  the  laws  oj  Utalth. 

Steele's  Physiology. 

See  page  33.  


ASTRONOMY. 

Willard's  School  Astronomy. 

By  means  of  clear  and  attractive  illustrations,  addressing  the  eye  in  many  cases  by 
analogies,  careful  definitions  of  all  necessary  technical  terms,  a  careful  avoidance  of  ver- 
biage and  unimportant  matter,  particular  attention  to  analysis,  and  a  general  adoption 
of  the  simplest  methods,  Mrs.  Willard  has  made  the  hest  and  most  attractive  elemen- 
Astronomy  extant. 

Mclntyre's  Astronomy  and  the  Globes. 

A  complete  treatise  for  intermediate  classes.     Highly  approved. 

Bartlett's  Spherical  Astronomy. 

The  West  Point  Course,  for  advanced  classes,  with  applications  to  the  current  wants 
of  Navigation,  Geography,  and  Chronology. 

Steele's  Astronomy. 

See  page  33.  

NATURAL    HISTORY. 

Carll's  Child's  Book  of  Natural  History. 

Illustrating  the  animal,  vegetable,  and  mineral  kingdoms,  with  application  to  the 
arts.     For  beginners.     BeautifuJlj  and  copiously  illustrated 

Anatomical  Technology.     Wilder  &  Gage. 

As  applied  to  the  domestic  cat.     For  the  use  of  students  of  medicine. 


ZOOLOGY. 

Chambers's  Elements  of  Zoology. 

A  complete  and  comprehensive  system  of  Zoology,  adapted  for  academic  instruction, 
presenting  a  systematic  view  of  the  animal  kingdom  as  a  portion  of  external  nature. 

Steele's  Zoology. 

See  page  33. 

38 


THE  NATIONAL    SERIES   OF   STANDARD   SCHOOL-BOOKS. 


LITERATURE. 


Gilman's  First  Steps  in  English  Literature. 

The  character  and  plan  of  this  exquisite  little,  text-book  may  be  best  understooa  nom 
an  analysis  of  its  contents  :  Introduction.  Historical  Period  of  Immature  English, 
with  Chart ;  Definition  of  Terms  ;  Languages  of  Europe,  with  Chart ;  Period  of  Mature 
English,  with  Chart ;  a  Chart  of  Bible  Translations,  a  Bibliography  or  Guide  to  General 
Reading,  and  other  aids  to  the  student. 

Cleveland's  Compendiums.     3  vols.     12mo. 

English  Literature.  American  Literature. 

English  Literature  of  the  XIXth  Century. 

In  these  volumes  are  gathered  the  cream  of  the  literature,  of  the  English-speaking 
people  for  the  school-room  and  the  general  reader.  Their  reputation  is  national.  More 
than  125,000  copies  have  been  sold. 

Boyd's  English  Classics.     6  vols.    Cloth.    12mo. 
Milton's  Paradise  Lost.  Thomson's  Seasons 

Young's  Night  Thoughts.  Pollok's  Course  of  Time. 

Cowpers  Task,  Table  Talk,  &c.    Lord  Bacon's  Essays. 
This  series  of  annotated   editions   of  great  English  writers  in  prose  and  poetry  is 

designed  for  critical  reading  and  parsing  in  schools.     Prof.  J.  R.  Boyd  proves  himself 

an  editor  of  high  capacity,  and  the  works  themselves  need  no  encomium.     As  auxiliary 

to  the  study  of  belles-lettres,  &c,  these  works  have  no  equal. 

Pope's  Essay  on  Man.     16mo.  Paper. 
Pope's  Homer's  Iliad.     32mo.  Roan. 

The  metrical  translation  of  the  great  poet  of  antiquity,  and  the  matchless  "  Essay  on 
the  Nature  and  State  of  Man,"  by  Alexander  Pope,  afford  superior  exercise  in  literature 
and  parsing. 


POLITICAL    ECONOMY. 

Champlin's  Lessons  on  Political  Economy. 

An  improvement  on  previous  treatises,  being  Bhorter,    yet   containing  everything 
essential,   with  a  view  of  recent  questions   in  finance,  &c.,  which   is   not  elsewhere 


found. 


39 


THE  NATIONAL   SERIES   OF  STANDARD   SCHOOL-BOOKS. 


AESTHETICS. 

Huntington's  Manual  of  the  Fine  Arts0 

A  view  of  the  rise  and  progress  of  art  in  different  countries,  p  brief  account  of  the 
moel  eminent  masters  of  art,  and  an  analysis  of  the  principles  01  art.  It  is  complete 
in  itself,  or  may  precede  to  advantage  the  critical  work  of  Lord  Karnes. 

Boyd's  Karnes's  Elements  of  Criticism. 

The  feest  edition  of  this  standard  work ;  without  the  study  of  which  none  maybe 
considered  proficient  in  the  science  of  the  perceptions.  No  other  study  can  be  pursued 
with  so  marked  an  effect  upon  the  taste  and  refinement  of  the  pupiL 


ELOCUTION. 

Watson's   Practical   Elocution. 

A  scientific  presentment  of  accepted  principles  of  elocutionary  drill,  with  black- 
board diagrams  and  full  collection  of  examples  for  class  drill.     Cloth.     90  pages,  12mo. 

Taverner  Graham's   Reasonable   Elocution. 

13a -ed  upon  the  belief  that  true  elocution  is  the  right  interpretation  of  thought, 
and  guiding  the  student  to  an  intelligent  appreciation,  instead  of  a  merely  mechanical 

knowledge,  of  its  rules. 

Zachos's  Analytic   Elocution. 

All  departments  of  elocution  —  such  as  the  analysis  of  the  voice  and  the  sentence, 
phonology,  rhythm,  expression,  gesture,  &c.  —  are  here  arranged  for  instruction  in 
classes,  illustrated  by  copious  examples. 


SPEAKERS. 

Northend's  Little  Orator. 
Northend's  Child's  Speaker. 

Two  little  works  of  the  same  grade  but  different  selections,  containing  simple   and 
attractive  pieces  for  children  under  twelve  years  of  age. 

Northend's  Young  Declaimer. 
Northend's  National  Orator. 

Two   volumes   of  prose,  poetry,  and   dialogue,  adapted  to  intermediate  and  grammar 
classes  respectively. 

Northend's  Entertaining  Dialogues. 

Extracts  eminently  adapted  to  cultivate  the  dramatic  faculties,  as  well  as  entertain. 

Oakey's  Dialogues  and  Conversations. 

For  school  exercises  and  exhibitions,  combining  useful  instruction. 
James's  Southern  Selections,  for  Reading  and  Oratory. 

Embracing  exclusively  Southern  literature 

Swett's  Common  School  Speaker. 
Raymond's  Patriotic  Speaker. 

A   superb  compilation   of  modern  eloquence   and  poetry,  with   original  dramatic 
exercises.    Nearly  every  eminent  modern  orator  is  represented 

40 


THE   NATIONAL    SERIES   OF   STANDARD    SCHOOL-BOOKS. 


MODERN    LANGUAGES. 


A    COMPLETE    COURSE   IN    THE    GERMAN. 

By  James  H.  Worman,  A.M.,  Professor  of  Modern  Languages  in  the  Adelphi  Acad- 
emy, Brooklyn,  L.  I. 

Worman's  First  German   Book. 
Worman's  Second   German  Book. 
Worman's   Elementary   German   Grammar. 
Worman's   Complete   German   Grammar. 

These  volumes  are  designed  for  intermediate  and  advanced  classes  respectively. 

Though  following  the  same  general  method  with  "  Otto  "  (that  of  "  Gaspey  ''),  our 
author  difi'ers  essentially  in  its  application.  He  is  more  practical,  more  systematic 
more  accurate,  and  besides  introduces  a  number  of  invaluable  features  which  have 
never  before  been  combined  in  a  German  grammar. 

Among  other  things,  it  may  be  claimed  for  Professor  Worman  that  he  has  been  the 
first  to  introduce,  in  an  American  text-book  for  learning  German,  a  system  of  analogy  and 
comparison  with  other  languages.  Our  best  teachers  are  also  enthusiastic  about  bis 
methods  of  inculcating  the  art  of  speaking,  of  understanding  the  spoken  language,  of 
correct  pronunciation  ;  the  sensible  and  convenient  original  classification  of  nouns  (in 
four  declensions),  and  of  irregular  verbs,  also  deserves  much  praise.  We  also  note  the 
use  of  heavy  type  to  indicate  etymological  changes  in  the  paradigms  and,  in  the  exer- 
cises, the  parts  which  specially  illustrate  preceding  rules. 

Worman's  Elementary  German   Reader. 
Worman's   Collegiate   German   Reader. 

The  finest  and  most  judicious  compilation  of  classical  and  standard  German  literature. 
These  works  embrace,  progressively  arranged,  selections  from  the  masterpieces  of 
Goethe,  Schiller,  Korner,  Seume,  Uhland,  Freiligrath,  Heine,  Schlegel,  Holty,  Lenau, 
Wieland,  Herder,  Lessing,  Kant,  Fichte,  Schelling,  Winkelmann,  Humboldt,  Xianke, 
Raumer,  Menzel,  Gervinus,  &c.,  and  contain  complete  Goethe's  "  Iphigenie,"  Schiller's 
"Jungfrau;"  also,  for  instruction  in  modern  conversational  German,  Beiiedix's 
"  Eigensinn." 

There  are,  besides,  biographical  sketches  of  each  author  contributing,  notes,  explan- 
atory and  philological  (after  the  text),  grammatical  references  to  all  leading  grammars, 
as  well  as  the  editor's  own,  and  an  adequate  Vocabulary. 

Worman's  German  Echo. 

Worman's   German   Copy-Books,   3  Numbers. 

On  the   same  plan  as  the  most  approved  systems  for  English  penmanship,  with 
progressive  copies. 

CHaUTAUQUA    SERIES. 
First  and  Second  Books  in  German. 

By  the  natural  or  Pestalozzian  System,  for  teaching  the  language  without  the  help 
of  the  Learner's  Vernacular.     By  James  II.  Worruan,  A.  M. 

These  books  belong  to  the  new  Chautauqua  German  Language  Series,  and  are  in- 
tended for  beginners  learning  to  speak  German.  The  peculiar  features  of  its  method 
are  :  — 

1.  It  teaches  the  language  by  direct  appeal  to  illustrations  of  the  objects 
referred  to,  and  dues  not  allow  the  student  to  guess  what  is  said,  lie  speaks  from  the 
first  hour  anderstandinglv  and  accurately.    Therefore, 

2.  Grammar  is  taught  both  analytically  and  synthetically  throughout  the 
course.  The  beginning  is  made  with  the  auxiliaries  of  tense  and  mood,  because  their 
kinship  with  the  English  makes  them  easily  intelligible  ;  then  follow  the  declensions  of 
nouns,  articles,  and  other  parts  of  speech,  always  systematically  arranged  It  is  easy 
to  confuse  the  pupil  by  giving  him  one  person  or  one  case  at  a  time.  This  pernicious 
practice  is  discarded.  Books  that  beget  unsystematic  habits  of  thought  are  worse  than 
wortldess. 

43 


THE   NATIONAL    SERIES   OF   STANDARD   SCHOOL-BOOKS. 


FRENCH. 

Worman's  First  Book  in  French. 

The  first  book  in  the  companion  series  to  the  successful  German  Series  by  the  same 
author,  and  intended  for  those  wishing  to  speak  French.  The  peculiar  features  of  Pro 
fessor  Wormau's  new  method  are  :  — 

1.  The  French  language  is  taught  without  the  help  of  English. 

2.  It  appeals  to  pictorial  illustrations  fin-  the  names  of  objects. 

3.  The  learner  speaks  from  the  first  hour  understandingly. 

4.  Grammar  is  taught  to  prevent  missteps  in  composition. 

5.  The  laws  of  ihe  language  are  taught  analytically  to  make  them  the  learner's  own 

inferences  (=  deductions). 

6.  Rapidity  of  progress  by  dependence  upon  association  and  contrasts. 

7.  Strictly  graded  lessons  and  conversations  on  familiar,  interesting,  and  instructive 

topics,  providing  the  words  and  idioms  of  every-day  life. 

8.  Paradigms  to  give  a  systematic  treatment   to  variable  inflections. 

9.  Heavy  type  for  inflections,  to  make  the  eye  a  help  to  the  mind 

10.     Hair  line  type  for  the  silent  Letters,  and  links  for  words  to  be  connected,  in  order 
to  teach  an  accurate  pronunciation. 

Worman's  French  Echo. 

This  is  not  a  mass  of  meaningless  and  parrot-like  phrases  thrown  together  for 
a  tourist's  use,  to  bewilder  him  when  in  the  presence  of  a  Frenchman. 

The  "  Echo  de  Paris  "  is  a  strictly  progressive  conversational  book,  beginning  with  sim- 
ple phrases  aud  leading  by  frequent  repetition  to  a  mastery  of  the  idioms  and  of  th-e 
every-day  language  used  in  business,  on  travel,  at  a  hotel,  in  the  chit-chat  of 
society. 

It  presupposes  an  elementary  knowledge  of  the  language,  such  as  may  be  acquired 
from  the  First  French  Book  by  Professor  Worman,  and  furnishes  a  running  French 
text,  allowing  the  learner  of  course  to  find  the  meaning  of  the  words  (in  the  appended 
Vocabulary),  and  forcing  him,  by  the  absence  of  English  in  the  text,  to  think  in 
French. 


Cher  Monsieur  Worman,  —  Vous  me 
demandezmon  opinion  sur  votre  "Echode 
Paris "  et  quel  usage  j'en  fais.  Je  ne 
saurais  mieux  vous  repondre  qu'en  repro- 
duisant  une  lettre  que  j'ecrivais  derniere- 
ment  a  un  eollegue  qui  etait,  me  disait-il, 
"  bien  fatigue  de  ces  insipides  livres  de 
dialogues." 

"  Vous  ne  connaissez  done  pas."  lui 
disais-je,  "TEchode  Paris,'  edite  par  le 
>r  Worman?  C'est  un  veritable 
tresor,  merveilleusement  sdapte  au  devel- 
oppement  de  la  conversation  familiere  et 
pratique,  telle  qu'on  la  vent  aujourd'hui. 

silent  livre  met  successive!] 
scene,  d'une  maniere  vive  et  inter 


toutes  les  circonstances  possibles  de  la  vie 
ordinaire.  Voyez  l'immense  avantage 
il  vous  transporte  en  Frame  ;  do  premier 
mot,  je  m'imagine,  et  mes  eleves  avec  moi, 
que  nous  sommes  a  Paris,  dans  la  rue,  sur 
une  place,  dans  une  gare,dans  un  salon, 
dans  une  chambiv,  voire  meine  a  Is  cui- 
sine ;  je  parte  comme  avec  des  Prancais  ; 
les  eleves  ne  songent  pas  a  tradnire  de 
1'anglais  pour  me  repondre  ;  ils  pensent 
en  franc. us  ;  ils  sont  Francais  pour  le 
moment  paries  yeux.  par  l'oreille.  par  la 
pens,  e  Quel  autre  livre  pourrait  produire 
cette  illusion  ?  .   .  ." 

Votre  tout  uevoue, 

A.   DE    KOUCEMONT. 


Illustrated  Language  Primers. 

French  and  English.  German  and  English. 

Spanish  and  English. 

The  names  of  common  objects  properly  illustrated  and  arranged  in  easy  lessons. 

Pujol's  Complete  French  Class-Book. 

Offers  in  one  volume,  methodically  arranged,  a  complete  French  course  —  usually 
embraced  in  series  of  from  five  x>>  twelve  books,  including  the  bulky  and  expensive 
lexicon.  Here  are  grammar,  conversation,  and  choice  literature,  selected  from  the 
best  French  authors.      Each   branch   is   thoroughly  handled  ;  and   the  student,   having 

diligently  completed  the  course  as  prescribed,  may  consider  himself,  without  further 
application,  au  fait  in  the  most  polite  and  elegant  language  of  modern  times. 

45 


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